Order and Hyper-order of Solutions of Second Order Linear Differential Equations

Abstract

We show that all non-trivial solutions of complex differential equation $f''+ A(z)f'+B(z)f = 0$ are of infinite order if coefficients A(z) and B(z) are of special type and establish a relation between the hyper-order of these solutions and the orders of coefficients A(z) and B(z). We have also extended these results to higher order complex differential equations.

1 Introduction

For two entire functions A(z) and $B(z)(\not \equiv 0)$, it is well known that all solutions of the second order linear differential equation

$$\begin{aligned} f''+A(z)f'+B(z)f=0 \end{aligned}$$

(1)

are entire functions. A necessary and sufficient condition for all the solutions of Eq. (1) to be of finite order is that the coefficients A(z) and B(z) are polynomials [14, Theorem 4.1]. It is easy to conclude that if any of the coefficients is a transcendental entire function then almost all solutions of Eq. (1) are of infinite order. The order of growth, lower order of growth, hyper-order of growth and exponent of convergence of nonzero zeros of an entire function f(z) are denoted by $\rho (f), \mu (f), \rho _2(f)$ and $\lambda (f)$, respectively. Our results are motivated by the following question asked by Gundersen:

Question 1

[9] If A(z) is a transcendental entire function such that $\lambda (A)<\rho (A)<\infty$ and if B(z) is a non-constant polynomial then does every non-trivial solution of Eq. (1) satisfy $\rho (f)=\infty ?$

However, there is a necessary condition for Eq. (1) to have a solution of finite order Gundersen [8, Theorem 2]. The implication of this result is that if Eq. (1) possesses a solution of finite order then $\rho (B)\le \rho (A)$. Therefore, if $\rho (A)<\rho (B)$, then all non-trivial solutions of Eq. (1) are of infinite order. It is already known that above condition is not sufficient, for example: $f''+e^{-z}f'+zf=0$ has all non-trivial solutions of infinite order, whereas $\rho (z)<\rho (e^{-z})$. It seems natural to ask what conditions A(z) and B(z) should have so that all non-trivial solutions of Eq. (1) are of infinite order. In this direction, many results are obtained. Gundersen [8] and Hellerstein et al. [10] proved that if $\rho (B)<\rho (A)\le 1/2$ or A(z) is a transcendental entire function with $\rho (A)=0$ and B(z) is a polynomial then f is of infinite order. Frei [4], Ozawa [20], Amemiya and Ozawa [1], Gundersen [6] and Langley [16] proved that all non-trivial solutions are of infinite order for the differential equation

$$\begin{aligned} f''+Ce^{-z}f'+B(z)f=0 \end{aligned}$$

for any nonzero constant C and for any non-constant polynomial B(z). There are many conditions on coefficients A(z) and B(z) so that all non-trivial solutions of Eq. (1) are of infinite order [19, 22, 23]. In this context, the authors [11] have proved that if B(z) is a transcendental entire function in Question 1 such that coefficients of Eq. (1) are of different order then conclusion of Question 1 holds true. However, a partial result on Question 1 is provided by Long et al. [17].

Our aim is to determine conditions on transcendental entire function B(z) of Eq. (1) so that when $\rho (A) = \rho (B)$, the conclusion of Question 1 holds true. Moreover, we know that the concept of hyper-order is an important ingredient in the study of infinite order solutions of complex differential equations. Therefore, we have also presented hyper-order of solutions of differential Eq. (1). Here is our first result in this direction:

Theorem 1

If A(z) is an entire function satisfying $\lambda (A)<\rho (A)<\infty$ and B(z) is a transcendental entire function with $\rho (A)\ne \mu (B)$, then all non-trivial solutions f of Eq. (1) satisfy:

• (1) $\rho (f)=\infty$, that is, all non-trivial solutions of Eq. (1) are of infinite order,

• (2) $\rho _2(f)=\max \{ \rho (A),\mu (B)\}$, whenever $\max \{ \rho (A),\mu (B)\}$ is a finite quantity.

Corollary 1

If B(z) is a transcendental entire function such that $\mu (A)\ne \mu (B)$ in the hypothesis of Theorem  1, then all non-trivial solutions of Eq. (1) are of infinite order.

It is interesting to note here that Theorem 1 and Corollary 1 holds for many entire functions B(z) whose lower order and order are different. One can find examples in Goldberg and Ostroviskii [5, p. 238]. We need to recall some notions for the statement of our next result:

Definition 1

Let $P(z)=a_{n}z^n+a_{n-1}z^{n-1}+\cdots +a_0$ be a polynomial of degree $n\in \mathbb {N}$ and $\delta (P,\theta )=\mathfrak {R}(a_ne^{\iota n \theta })$. A ray $\gamma = re^{\iota \theta }$ is called a critical ray of $e^{P(z)}$ if $\delta (P,\theta )=0.$

The rays $\arg {z}=\theta$ such that $\delta (P,\theta )=0$ divides the complex plane into 2n sectors of equal length $\pi /n$. Also, $\delta (P,\theta )>0$ and $\delta (P,\theta )<0$ in the alternative sectors. Suppose that $0\le \phi _1<\theta _1<\phi _2<\theta _2<\cdots<\phi _n<\theta _n<\phi _{n+1}=\phi _1+2\pi$ are 2n critical rays of $e^{P(z)}$ satisfying $\delta (P,\theta )>0$ for $\phi _i<\theta <\theta _i$ and $\delta (P,\theta )<0$ for $\theta _i<\theta <\phi _{i+1}$ where $i=1,2,3, \ldots , n$. Now, we fix some notations:

$$\begin{aligned} E^+ = \{ \theta \in [0,2\pi ]: \delta (P,\theta )\ge 0\},\quad E^- = \{ \theta \in [0,2\pi ]: \delta (P,\theta )\le 0 \}. \end{aligned}$$

For $\alpha$ and $\beta$ with $0\le \alpha <\beta \le 2\pi$, let

$$\begin{aligned} \varOmega (\alpha ,\beta )= & {} \{z\in \mathbb {C}: \alpha<\arg z<\beta \}\\ \varOmega (\alpha ,\beta ,r)= & {} \{z\in \mathbb {C}: |z|<r, \alpha<\arg z<\beta \}\\ \varOmega (\alpha ,\beta ,r,\infty )= & {} \{z\in \mathbb {C}: r<|z|<\infty , \alpha<\arg z <\beta \}. \end{aligned}$$

Example 1

The function $e^z$ has two critical rays namely $\theta =-\pi /2$ and $\theta = \pi /2$. Also, $E^+=[-\pi /2,\pi /2]$ and $E^-=[\pi /2,3\pi /2]$.

The following are definitions of Borel direction and function extremal to Yang’s inequality with examples.

Definition 2

[24] For a meromorphic function B(z) of order $\rho (B)\in (0,\infty )$ in the finite plane, the ray $\arg {z}=\theta _0$ is called Borel direction of B(z) of order $\rho (B)$ if for any $\epsilon >0$, the equality

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log {n(\varOmega (\theta _0-\epsilon ,\theta _0+\epsilon ,r), B=a)}}{\log {r}}=\rho (B) \end{aligned}$$

holds for every $a\in \mathbb {C}\cup \{\infty \}$, with at most two possible exceptions, where $n(\varOmega (\theta _0-\epsilon ,\theta _0+\epsilon , r),B=a)$ denotes the number of zeros, counting with the multiplicities, of the function $B(z)-a$ in the region $\varOmega (\theta _0-\epsilon ,\theta _0+\epsilon ,r)$. When $a=\infty$, then $B(z)=a$ is replaced by 1/B(z).

Example 2

The entire function $B(z)=e^{z}$ has two Borel directions namely $\pi /2$ and $-\pi /2$.

Definition 3

If B(z) is an entire function with $\rho (B)\in (0,\infty )$ having p number of finite deficient values and q number of Borel direction, then B(z) is said to be extremal to Yang’s inequality if $p=q/2.$

Remark 1

[21] If B(z) is an entire function extremal to Yang’s inequality, then $\rho (B)=\mu (B).$

Example 3

[24, p. 210] Consider the entire function $B(z)=\int _{0}^{z}e^{-t^n}\mathrm{d}t$ of order n. It has n number of finite deficient values equal to

$$\begin{aligned} a_k=e^{\frac{i2\pi k}{n}} \int _{0}^{\infty }e^{-t^n}\mathrm{d}t, \quad k=0,1,2,\ldots , n-1 \end{aligned}$$

and 2n Borel directions equal to

$$\begin{aligned} \Phi _i=\frac{(2i-1)\pi }{2n},\quad i=0,1,2,\ldots , 2n-1. \end{aligned}$$

Since $p=q/2$, this function is extremal to Yang’s inequality.

We now state a conjecture introduced by Denjoy [3] which gives a relation between the order of an entire function and its finite asymptotic values.

Denjoy’s Conjecture If B(z) is an entire function of finite order and having p distinct finite asymptotic values, then $p\le 2\rho (B)$.

Definition 4

An entire function B(z) is said to be extremal to Denjoy’s conjecture when $p= 2\rho (B)$.

Example 4

[26, p. 210] Let

$$\begin{aligned} B(z)=\int _{0}^{z} \frac{\mathrm{sin}{(t^p)}}{t^p}\mathrm{d}t \end{aligned}$$

where $p\in \mathbb {N}$. Then, $\rho (B)=p$ and B(z) has 2p distinct finite asymptotic values, namely

$$\begin{aligned} a_j=e^{\frac{j\pi \iota }{p}}\int _{0}^{\infty }\frac{\mathrm{sin}{(r^p)}}{r^p}\mathrm{d}r \end{aligned}$$

for $j=1,2,\ldots , 2p$.

We are now prepared to state our next result which gives conditions so that whenever $\rho (A)=\rho (B)$ then all non-trivial solutions of Eq. (1) possess infinite order and finite hyper-order.

Theorem 2

If $A(z)=v(z)e^{P(z)}$ is an entire function, where P(z) is a non-constant polynomial and $v(z) (\not \equiv 0)$ is an entire function with $\rho (v)<\rho (A)$ and B(z) is a transcendental entire function satisfying either of the following conditions:

• (a) B(z) is an entire function extremal to Yang’s inequality such that no Borel direction of B(z) coincides with any of the critical rays of $e^{P(z)}$,

• (b) B(z) is an entire function extremal to Denjoy’s conjecture,

  then all non-trivial solutions f of Eq. (1) satisfy:

  • 1. $\rho (f)=\infty$, that is, all non-trivial solutions of Eq. (1) are of infinite order.

  • 2. $\rho _2(f)=\max \{ \rho (A),\rho (B)\} .$

As illustrations, we see that the differential equations

$$\begin{aligned}&f''+e^{\iota z}f'+e^zf=0 \end{aligned}$$

(2)

$$\begin{aligned}&f''+\left( \sin {z}\right) e^{z^3}f'+\left( \int _{0}^{z} t^{-3} \sin {t^3}\mathrm{d}t\right) f=0 \end{aligned}$$

(3)

have all non-trivial solutions of infinite order, respectively, by part (a) and part (b) of Theorem 2. The conclusion for the differential Eq. (2) also follows from [8, Theorem 4]. In differential Eq. (3), the function $B(z)=\int _{0}^{z} t^{-3} \sin {t^3}\mathrm{d}t$ comes from Example 4. Part (a) and part (b) of Theorem 2 are motivated, respectively, by [19, Theorem 1.3] and [22, Theorem 1.6].

Here, we give two differential equations to justify that conditions in Theorem 2 cannot be removed.

The finite order function $f(z)=z$ satisfies $f''+ze^zf'-e^zf=0$ and $f(z)=e^{-z}$ satisfies $f''+e^zf'+(e^z-1)f=0$ where hypothesis of Theorem 2(a) and (b), respectively, is not satisfied.

2 Preliminary Results

This section includes some standard definitions and known basic results. For a set $G\subset (0,\infty )$, the linear measure and for a set $S\subset (1,\infty )$ the logarithmic measure, upper logarithmic density and lower logarithmic density are defined as

$$\begin{aligned} m(G)= & {} \int _{G}\mathrm{d}t, \quad m_l(S)=\int _{S}\frac{1}{t}\mathrm{d}t \\ \overline{\text {log dens}}(S)= & {} \limsup _{r\rightarrow \infty }\frac{m_l\left( S\cap [1,r]\right) }{\log r}\quad \text{ and }\quad \underline{\text {log dens}}(S)=\liminf _{r\rightarrow \infty }\frac{m_l\left( S\cap [1,r]\right) }{\log r} \end{aligned}$$

, respectively, and $0\le \underline{\text {log dens}}(S)\le \overline{\text {log dens}}(S)\le 1$. The following lemma of Gundersen [7] provides the estimates for transcendental meromorphic function.

Lemma 1

Let f be a transcendental meromorphic function and $\varGamma = \{ (k_1,j_1), \ldots ,(k_m,j_m) \}$ denote finite set of distinct pairs of integers that satisfy $k_i > j_ i \ge 0$ for $i=1,2, \ldots ,m$. If $\alpha >1$ and $\epsilon >0$ be given real constants, then the following statements holds:

• (i) there exists a set $E_1 \subset [0,2\pi )$ that has linear measure zero and there exists a constant $c>0$ that depends only on $\alpha$ and $\varGamma$ such that if $\psi _0 \in [0,2\pi )\setminus E_1,$ then there is a constant $R_0(\psi _0)>0$ so that for all z satisfying $\arg z =\psi _0$ and $|z| \ge R_0$ and for all $(k,j)\in \varGamma$, we have

  $$\begin{aligned} \left| \frac{f^{(k)}(z)}{f^{(j)}(z)}\right| \le c \left( \frac{T(\alpha r,f)}{r} \log ^{\alpha }{r} \log {T(\alpha r,f)} \right) ^{k-j}. \end{aligned}$$

  (4)

  If f is of finite order, then f satisfies:

  $$\begin{aligned} \left| \frac{f^{(k)}(z)}{f^{(j)}(z)}\right| \le |z|^{(k-j)(\rho (f)-1+\epsilon )} \end{aligned}$$

  (5)

  for all z satisfying $\arg z =\psi _0\notin E_1$, $|z| \ge R_0$ and for all $(k,j)\in \varGamma$.

• (ii) there exists a set $E_2\subset (1,\infty )$ that has finite logarithmic measure and there exists a constant $c>0$ that depends only on $\alpha$ and $\varGamma$ such that for all z satisfying $|z|=r\notin E_2\cup [0,1]$ and for all $(k,j)\in \varGamma$, inequality (4) holds. If f is of finite order, then f satisfies inequality (5), for all z satisfying $|z| \not \in E_2\cup [0,1]$ and for all $(k,j)\in \varGamma$.

The following lemma provides estimates for an entire function A(z) outside a set of linear measure zero.

Lemma 2

[2] Let $A(z)=v(z)e^{P(z)}$ be an entire function, where P(z) is a non-constant polynomial of degree n and $v(z) (\not \equiv 0)$ is an entire function satisfying $\rho (v)<n$. Then, for every $\epsilon >0$ there exists $E \subset [0,2\pi )$ of linear measure zero such that

• (i) for $\theta \in E^+\setminus E$, there exists $R=R(\theta )>1$ such that

  $$\begin{aligned} |A(re^{\iota \theta })| \ge \exp \{ (1-\epsilon ) \delta (P,\theta )r^n \} \end{aligned}$$

  (6)

  for $r>R.$

• (ii) for $\theta \in E^-\setminus E$, there exists $R=R(\theta )>1$ such that

  $$\begin{aligned} |A(re^{\iota \theta })| \le \exp \{(1-\epsilon )\delta (P,\theta ) r^n \} \end{aligned}$$

  (7)

  for $r>R.$

Using Residue theorem, Kwon [13] proved the following lemma:

Lemma 3

If f is a non-constant entire function, then there exist a real number $R>0$ such that for all $r\ge R$ there exists z with $|z|=r$ satisfying

$$\begin{aligned} \left| \frac{f(z)}{f'(z)} \right| \le r. \end{aligned}$$

(8)

Kwon [13] found the lower bound for the hyper-order of all non-trivial solutions f as given in the following result:

Lemma 4

[13] If A(z) and B(z) are entire functions such that $\rho (A)<\rho (B)$, then

$$\begin{aligned} \rho _2(f)\ge \rho (B) \end{aligned}$$

for all non-trivial solutions f of Eq. (1).

The following lemma gives the definition of hyper-order of an entire function f with infinite order using the central index of f.

Lemma 5

[28] Let f be an entire function with infinite order and v(r, f) denote the central index of f, then

$$\begin{aligned} \rho _2(f)=\limsup _{r\rightarrow \infty }\frac{\log \log v(r,f)}{\log r}. \end{aligned}$$

Zongxuan [27] provided upper bound for hyper-order of solutions f of Eq. (1).

Lemma 6

If A(z) and B(z) are entire functions of finite order, then

$$\begin{aligned} \rho _2(f)\le \max \{ \rho (A), \rho (B)\} \end{aligned}$$

for all solutions f of Eq. (1).

The following three lemmas describe properties of an entire function with finite lower order.

Lemma 7

[23] If B(z) is an entire function with $\mu (B)\in [0,1)$. Then, for every $\alpha \in (\mu (B),1)$, there exists a set $S\subset [1,\infty )$ such that

$$\begin{aligned} \overline{\text {log dens}}(S)\ge 1-\frac{\mu (B)}{\alpha } \,\mathrm{and }\, m(r)>M(r)\cos {\pi \alpha } \end{aligned}$$

for all $r\in S$, where $m(r)=\inf _{|z|=r}\log |B(z)|$ and $M(r)=\sup _{|z|=r} \log |B(z)|$.

The above lemma is also true for an entire function B(z) with $\rho (B)<1/2.$ We can get next lemma easily using Lemma 7.

Lemma 8

[23] Suppose that B(z) is an entire function with $\mu (B)\in (0,1/2)$. Then, for any $\epsilon >0$, there exists $(r_n)\rightarrow \infty$ such that

$$\begin{aligned} |B(r_ne^{\iota \theta })|>\exp \{r_n^{\mu (B)-\epsilon }\} \end{aligned}$$

for all $\theta \in [0,2\pi )$.

Lemma 9

[22] If B(z) is an entire function with $\mu (B)\in \big [1/2,\infty )$, then there exists a sector $\varOmega (\alpha , \beta ),$ $\beta -\alpha \ge \pi /\mu (B)$, such that

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log \log |B(re^{\iota \theta })|}{\log r}\ge \mu (B) \end{aligned}$$

for all $\theta \in \varOmega (\alpha ,\beta )$, where $0\le \alpha <\beta \le 2 \pi .$

Here, we fix notations for Borel directions of an entire function B(z) which is extremal to Yang’s inequality. The rays $\arg {z}=\Phi _i$, $i=1,2,3,\ldots ,q$ denote Borel directions of the function B(z) such that $0\le \Phi _1<\Phi _2<\cdots<\Phi _q<\Phi _{q+1}=\Phi _1+2\pi$. The following two lemmas give property of function extremal to Yang’s inequality.

Lemma 10

[21] If B(z) is an entire function extremal to Yang’s inequality and $b_i$, $i=1,2,3,\ldots , q/2$ are the deficient values of B(z). Then, for each $b_i$, $i=1,2,3,\ldots ,q/2$, there exists a corresponding sector $\varOmega (\Phi _i,\Phi _{i+1})$ such that for every $\epsilon >0$

$$\begin{aligned} \log {\frac{1}{|B(z)-b_i|}}>C(\Phi _i,\Phi _{i+1}, \epsilon , \delta (b_i,B))T(|z|,B) \end{aligned}$$

(9)

holds for all $z\in \varOmega (\Phi _i+\epsilon ,\Phi _{i+1}-\epsilon , r,\infty )$, where $C(\Phi _i,\Phi _{i+1}, \epsilon , \delta (b_i,B))$ is a positive constant depending on $\Phi _i, \Phi _{i+1}, \epsilon$ and $\delta (b_i,B)$, where $\delta (\_,B)$ is deficient function of B(z).

Lemma 11

[19] Suppose that B(z) is an entire function extremal to Yang’s inequality and there exists $\arg {z}=\theta$ with $\Phi _i<\theta <\Phi _{i+1},$ $1\le i\le q$ such that

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log {\log {|B(re^{\iota \theta })|}}}{\log {r}}=\rho (B) \end{aligned}$$

(10)

then $\Phi _{i+1}-\Phi _i=\pi /\rho (B)$.

Next, lemma gives property of an entire function inside a sector $\varOmega (\psi _1,\psi _2)$.

Lemma 12

[25] Suppose that B(z) is an entire function with $0<\rho (B)<\infty$ and $\varOmega (\psi _1,\psi _2)$ be a sector with $\psi _2-\psi _1<\pi /\rho (B)$. If there exists a Borel direction $\arg {z}=\Phi$ in $\varOmega (\psi _1,\psi _2)$, then there exists at least one of the rays $\arg {z}=\psi _i$, $i=1$ or 2 such that

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log {\log {|B(re^{\iota \psi _i})|}}}{\log {r}}=\rho (B). \end{aligned}$$

(11)

Following lemma gives the property of an entire function extremal to Denjoy’s conjecture.

Lemma 13

[26] If B(z) is an entire function extremal to Denjoy’s conjecture, then for any $\theta \in (0,2\pi )$ either $\arg {z}=\theta$ is a Borel direction of B(z) or there exists a constant $\sigma \in (0,\pi /4)$ such that

$$\begin{aligned} \lim _{{|z|\rightarrow \infty }_{ z \in (\varOmega (\theta -\sigma ,\theta +\sigma )\setminus S)}} \frac{\log \log |B(z)|}{\log {|z|}} =\rho (B) \end{aligned}$$

(12)

where $S\subset \varOmega (\theta -\sigma ,\theta +\sigma )$ such that

$$\begin{aligned} \lim _{r\rightarrow \infty }m(\varOmega (\theta -\sigma ,\theta +\sigma , r, \infty )\cap S)=0. \end{aligned}$$

3 Proof of Main Results

3.1 Propositions

This subsection includes proofs of results which are used in our proof of main theorems.

We proved the following result which is motivated by [15, Lemma 4] and [18, Lemma 3.1]:

Proposition 1

If B(z) is an entire function with $\mu (B)<\rho (B)$, then for given $\epsilon >0$ there exists $S\subset (1,\infty )$ satisfying $\overline{\text {log dens}}(S)>0$ such that

$$\begin{aligned} \log {M(r,B)}\le r^{\mu (B)+\epsilon } \end{aligned}$$

(13)

for all sufficiently large $r\in S$.

Proof

We need to consider $\epsilon >0$ such that $\mu (B)<\mu (B)+\epsilon \le \rho (B)$. By the definition of lower order, for an arbitrary fixed $\epsilon >0$ we get

$$\begin{aligned} \log {M(r,B)}\le r^{\mu (B)+\epsilon } \end{aligned}$$

for arbitrary large r. Define $S=\{r>1:\log {M(r,B)}\le r^{\mu (B)+\epsilon }\}$ and $\overline{S}=[1,\infty )\setminus S$. Let us suppose that $\overline{\text {log dens}}(S)=0$. Choose $0<\alpha <1$ then using the definition of logarithmic upper density, for $0<\beta <(1-\alpha )/(1+\alpha )$ there exist $r_0>1$ such that

$$\begin{aligned} \frac{m_l\left( S\cap [1,r]\right) }{\log {r}}<\beta \end{aligned}$$

for all $r>r_0$. From here, we will get

$$\begin{aligned} \frac{m_l\left( S\cap [r^{\alpha },r]\right) }{\log {r}}\le \frac{m_l\left( S\cap [1,r]\right) }{\log {r}}<\beta \end{aligned}$$

for all $r>r_0$. Consider

$$\begin{aligned} (1-\alpha )\log {r}&=\int _{r^{\alpha }}^{r}\frac{1}{t}\mathrm{d}t=m_l\left( S\cap [r^{\alpha },r]\right) +m_l\left( \overline{S}\cap [r^{\alpha },r]\right) \\&<\beta \log {r}+m_l\left( \overline{S}\cap [r^{\alpha },r]\right) <\frac{1-\alpha }{1+\alpha }\log {r}+m_l\left( \overline{S}\cap [r^{\alpha },r]\right) \\ \end{aligned}$$

for all $r>r_0$. This will imply

$$\begin{aligned} m_l\left( \overline{S}\cap [r^{\alpha },r]\right) >\frac{\alpha (1-\alpha )}{1+\alpha }\log {r} \end{aligned}$$

for all $r>r_0$. Therefore, there exists $s\in \overline{S}\cap [r^{\alpha },r]$ so that

$$\begin{aligned} r^{\alpha (\mu (B)+\epsilon )}\le s^{\mu (B)+\epsilon }<\log {M(s,B)}\le \log {M(r,B)} \end{aligned}$$

for all $r>r_0$, which implies $\alpha (\mu (B)+\epsilon )\le \mu (B)$. This is a contradiction as $0<\alpha <1$ is arbitrary and we get $\mu (B)+\epsilon \le \mu (B)$ as $\alpha \rightarrow 1$. $\square$

Remark 2

From the definition of lower order of an entire function B(z) we have, for each $\epsilon >0$ there exists $R>0$ such that

$$\begin{aligned} M(r,B)\ge \exp {\left( r^{\mu (B)-\epsilon }\right) } \end{aligned}$$

for all $r>R.$

The following result is motivated from Lemma 6 and provides an upper bound for hyper-order of solution of Eq. (1).

Proposition 2

If A(z) and B(z) are entire functions such that $\rho (A)$ and $\mu (B)$ are finite then

$$\begin{aligned} \rho _2(f)\le \max \{ \rho (A),\mu (B)\} \end{aligned}$$

(14)

for all solutions f of Eq. (1).

Proof

When order of f is finite then (14) follows easily. Therefore, let $\rho (f)=\infty$ and $\max \{ \rho (A),\mu (B)\} = \rho$. From the definition of order of function A(z), for each $\epsilon >0$ there exists $R>0$ such that

$$\begin{aligned} |A(re^{\iota \theta }|\le \exp {\left( r^{\rho +\epsilon }\right) } \end{aligned}$$

(15)

for all $r>R$. From Proposition 1, for $\epsilon > 0$ there are sets $S\in (1,\infty )$, with $\overline{\text {log dens}}(S)>0$ such that

$$\begin{aligned} |B(re^{\iota \theta }|\le \exp {\left( r^{\rho +\epsilon }\right) } \end{aligned}$$

(16)

for sufficiently large $r\in S$. From Theorem 3.2 of [14, p. 51], we choose z satisfying $|z|=r$ and $|f(z)|=M(r,f)$ then there exists a set $F\subset \mathbb {R_+}$ having finite logarithmic measure such that

$$\begin{aligned} \frac{f^{(m)}(z)}{f(z)}=\left( \frac{v(r,f)}{z} \right) ^m(1+o(1)) \end{aligned}$$

(17)

for $m=1,2$ and for all $|z|=r\notin F$, where v(r, f) is the central index of the function f. Also, $\overline{\text {log dens}}(S\setminus F)>0$, using $m_l(F)<\infty$. From Eqs. (1), (15), (16) and (17), we get

$$\begin{aligned} \left( \frac{v(r,f)}{|z|} \right) ^2|(1+o(1))|\le \exp { \left( r^{\rho +\epsilon }\right) } \left( \frac{v(r,f)}{|z|} \right) |(1+o(1))|+\exp {\left( r^{\rho +\epsilon }\right) } \end{aligned}$$

(18)

for all $|z|=r\in S\setminus F$, from here we get

$$\begin{aligned} \limsup _{r\rightarrow \infty } \frac{\log \log {v(r,f)}}{\log {r}}\le \rho +\epsilon . \end{aligned}$$

(19)

Using Lemma 5, we have $\rho _2(f)\le \rho$, as $\epsilon >0$ is arbitrary. $\square$

3.2 Proof of Part 1 of Theorem 1

Proof

If $\rho (A)<\mu (B)$, then result follows from [8, Theorem 2]. So, assume that $\mu (B)<\rho (A)$ and f is a non-trivial solution of Eq. (1) with finite order. Then, using part (i) of Lemma 1, for each $\epsilon >0,$ there exists a set $E_1 \subset [0,2\pi )$ that has linear measure zero, such that if $\psi _0 \in [0,2\pi )\setminus E_1,$ then there is a constant $R_0=R_0(\psi _0)>0$ such that

$$\begin{aligned} \left| \frac{f^{(k)}(z)}{f(z)}\right| \le |z|^{2\rho (f)}, \quad k=1,2 \end{aligned}$$

(20)

for all z satisfying $\arg z =\psi _0$ and $|z| \ge R_0$. Since $\lambda (A)<\rho (A)$, we have $A(z)=v(z)e^{P(z)}$, where P(z) is a non-constant polynomial of degree n and v(z) is an entire function such that $\rho (v)=\lambda (A)<\rho (A)$. Then, using Lemma 2, there exists $E\subset [0,2\pi )$ with linear measure zero such that for $\theta \in E^-\setminus \left( E\cup E_1\right)$ we have

$$\begin{aligned} |A(re^{\iota \theta })|\le \exp \left( (1-\epsilon ) \delta (P,\theta )r^n \right) \end{aligned}$$

(21)

for sufficiently large r.

We have following three cases on lower order of B(z):

Case 1 when $0<\mu (B)<1/2$, then from Lemma 8, there exists $(r_m)\rightarrow \infty$ such that

$$\begin{aligned} |B(r_me^{\iota \theta })|>\exp {\left( r_m^{\mu (B)-\epsilon } \right) } \end{aligned}$$

(22)

for all $\theta \in [0,2\pi )$ and for sufficiently large $r_m$. Using Eqs. (1), (20), (21) and (22), we have

$$\begin{aligned} \exp {\left( r_m^{\mu (B)-\epsilon }\right) }&<|B(r_me^{\iota \theta })|\le \frac{|f''(r_me^{\iota \theta })|}{|f(r_me^{\iota \theta })|}+|A(r_me^{\iota \theta })|\frac{|f'(r_me^{\iota \theta })|}{|f(r_me^{\iota \theta })|} \\&\le r_m^{2\rho (f)}\{ 1+\exp \left( (1-\epsilon ) \delta (P,\theta )r_m^n \right) \} \\&= r_m^{2\rho (f)} \{1+o(1) \} \end{aligned}$$

for all $\theta \in E^-\setminus (E \cup E_1)$ and for sufficiently large $r_m$. This will lead to a contradiction.

Case 2 Now, if $\mu (B)\ge 1/2$, then by Lemma 9 we have that there exists a sector $\varOmega (\alpha ,\beta )$, $0\le \alpha <\beta \le 2\pi$, $\beta -\alpha \ge \pi /\mu (B)$ such that

$$\begin{aligned} \limsup _{r\rightarrow \infty } \frac{\log \log |B(re^{\iota \theta }|}{\log r}\ge \mu (B) \end{aligned}$$

(23)

for all $\theta \in \varOmega (\alpha , \beta )$. Since $\mu (B)<\rho (A)$ therefore, there exists $\varOmega (\alpha ',\beta ') \subset \varOmega (\alpha ,\beta )$ such that for all $\phi \in \varOmega (\alpha ', \beta ')\setminus \left( E\cup E_1\right)$, Eq. (21) holds true. From Eq. (23), we get

$$\begin{aligned} \exp {\left( r^{\mu (B)-\epsilon }\right) }\le |B(re^{\iota \phi })| \end{aligned}$$

(24)

for $\phi \in \varOmega (\alpha ',\beta ')\setminus \left( E\cup E_1\right)$ and $r>R.$ As done in case 1, using Eqs. (1), (20), (21) and (24) we get contradiction for sufficiently large r.

Case 3 If $\mu (B)=0$, then from Lemma 7 for $\alpha \in (0,1)$, there exists a set $S\subset [1,\infty )$ with $\overline{\text {log dens}}(S)=1$ such that

$$\begin{aligned} m(r)>M(r)\cos {\pi \alpha } \end{aligned}$$

where $m(r)=\inf _{|z|=r}\log |B(z)|$ and $M(r)=\sup _{|z|=r} \log |B(z)|$. Then,

$$\begin{aligned} \log {|B(re^{\iota \theta })|}>\log {M(r,B)}^{\frac{1}{\sqrt{2}}} \end{aligned}$$

(25)

for all $\theta \in [0,2\pi )$ and $r\in S.$ Now, using Eqs. (1), (20), (21) and (25), we get

$$\begin{aligned} M(r,B)^{\frac{1}{\sqrt{2}}}<|B(re^{\iota \theta })|\le r^{2\rho (f)}\{1+\exp {(1-\epsilon )\delta (P,\theta )r^n} \} \end{aligned}$$

for $\theta \not \in E\cup E_1$, $\delta (P,\theta )<0$ and $r>R$, $r\in S$. This implies that

$$\begin{aligned} \limsup _{r\rightarrow \infty } \frac{\log M(r,B)}{\log r}<\infty \end{aligned}$$

which is impossible as B(z) is a transcendental entire function. Thus, a non-trivial solution f with finite order of Eq. (1) cannot exist in this case.

Therefore, all non-trivial solutions of Eq. (1) are of infinite order. $\square$

3.3 Proof of Part 2 of Theorem 1

Proof

We know that under the hypothesis of the theorem, all non-trivial solutions f of Eq. (1) are of infinite order. It follows from Lemma 1 that for $\epsilon >0$, there exists a set $E_2\subset [1,\infty )$ with finite logarithmic measure such that

$$\begin{aligned} \left| \frac{f''(z)}{f'(z)}\right| \le c[T(2r,f)]^2 \end{aligned}$$

(26)

for sufficiently large $|z|=r\notin E_2\cup [0,1]$ where $c>0$ is a constant.

Let $\rho (A)<\mu (B)$, then from Lemma 4 and Proposition 2, we get that $\rho _2(f)=\max \{ \rho (A),\mu (B)\}$, for all non-trivial solutions f of Eq. (1).

If $\mu (B)<\rho (A)$. It is easy to choose $\eta$ such that $\mu (B)<\eta <\rho (A)$. From Lemma 2, we have

$$\begin{aligned} \exp {\{ (1-\epsilon )\delta (P,\theta )r^n \} }\le |A(re^{\iota \theta }| \end{aligned}$$

(27)

for all $\theta \notin E$, $\delta (P,\theta )>0$ and for sufficiently large r.

Using Proposition 1, we have

$$\begin{aligned} M(r,B)\le \exp \{r^{\eta }\} \end{aligned}$$

(28)

for sufficiently large $r\in S$, where $\overline{\text {log dens}}(S\setminus E_2)>0$ as $m_l(E_2)<\infty$. Thus, from Eqs. (1), (8), (26), (27) and (28), we have

$$\begin{aligned} \exp {\{ (1-\epsilon )\delta (P,\theta )r^n\} }&\le |A(re^{\iota \theta }| \\&\le \left| \frac{f''(re^{\iota \theta })}{f'(re^{\iota \theta })}\right| +|B(re^{\iota \theta })|\left| \frac{f(re^{\iota \theta })}{f'(re^{\iota \theta })}\right| \\&\le c[T(2r,f)]^2+\exp {\left( r^{\eta }\right) }r \\&\le \mathrm{d}r\exp {\left( r^{\eta }\right) }[T(2r,f)]^2 \end{aligned}$$

for all $\theta \notin E$, $\delta (P,\theta )>0$ and for sufficiently large $r\in S\setminus E_2$. Since $\eta <\rho (A)=n$, we have

$$\begin{aligned} \exp {\{ (1-\epsilon )\delta (P,\theta )\} }\exp {\{ (1-o(1))r^n\} }\le \mathrm{d}r[T(2r,f)]^2 \end{aligned}$$

(29)

for sufficiently large $r\in S\setminus E_2$. Thus,

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log {\log {T(r,f)}}}{\log {r}}\ge n. \end{aligned}$$

(30)

Now, using Eq. (30) and Proposition 2, we get the desired result. $\square$

3.4 Proof of Part 1 of Theorem 2

Proof

If we consider coefficients A(z) and B(z) such that $\rho (A)\ne \rho (B)$, then result holds true [11, Theorem 4]. Therefore, it is sufficient to consider $\rho (A)=\rho (B)=n$ for some $n\in \mathbb {N}$.

• (a) Suppose B(z) is extremal to Yang’s inequality and there exists a non-trivial solution f of Eq. (1) of finite order, then using part (ii) of Lemma 1, for each $\epsilon >0,$ there exists a set $E_2 \subset (1,\infty )$ that has finite logarithmic measure, such that

  $$\begin{aligned} \left| \frac{f^{(k)}(z)}{f(z)}\right| \le |z|^{2\rho (f)}, \quad k=1,2 \end{aligned}$$

  (31)

  for all z satisfying $|z|\not \in E_2\cup [0,1]$. Since B(z) is extremal to Yang’s inequality, therefore there exists sectors $\varOmega _i(\Phi _i,\Phi _{i+1})$, $i=1,2,3,\ldots , q$ such that in alternative sectors either Eq. (9) or Eq. (10) holds for B(z). Let $\varOmega _1(\Phi _1,\Phi _2)$, $\varOmega _3(\Phi _3,\Phi _4)$, $\ldots , \varOmega _{2q-1}(\Phi _{2q-1},\Phi _{2q})$ be the sectors such that

  $$\begin{aligned} \log {\frac{1}{|B(z)-b_i|}}>C T(|z|,B) \end{aligned}$$

  (32)

  holds for all $z\in \varOmega _i(\Phi _i+\epsilon ,\Phi _{i+1}-\epsilon , r,\infty )$, where $C=C(\Phi _i,\Phi _{i+1}, \epsilon , \delta (b_i,B))$ is a positive constant. Also, suppose $\varOmega _2(\Phi _2,\Phi _3),$ $\varOmega _4(\Phi _4,\Phi _6)$, $\ldots , \varOmega _{2q}(\Phi _{2q},\Phi _{2q+1})$ are sectors for which there exists $re^{\iota \theta _{2i}}\in \varOmega _{2i}(\Phi _{2i},\Phi _{2i+1})$ such that

  $$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log {\log {|B(re^{\iota \theta _{2i}})|}}}{\log {r}}=n \end{aligned}$$

  (33)

  holds and $\Phi _{2i+1}-\Phi _{2i}=\pi /n$ where $i=1,2,\ldots , q$. Now, we have following two cases to discuss:

  Case 1:

  let us suppose that there is a Borel direction $\Phi$ of B(z) such that $\theta _i<\Phi <\phi _{i+1}$ for any $i=1,2,3, \ldots ,n$. Then, we can easily choose $\psi _1$ and $\psi _2$ such that $\theta _i<\psi _1<\Phi<\psi _2<\phi _{i+1}$. It is evident from Lemma 12 that without loss of generality we can choose $\psi _2$ such that

  $$\begin{aligned} \limsup _{r\rightarrow \infty } \frac{\log {\log {|B(re^{\iota \psi _2})|}}}{\log {r}}=n. \end{aligned}$$

  (34)

  Thus, for $r\notin E_2\cup [0,1]$ from Eqs. (1), (7), (31) and (34) we have

  $$\begin{aligned} \exp {\{ r^{n-\epsilon }\} }&\le |B(re^{\iota \psi _2})|\le \left| \frac{f''(re^{\iota \psi _2})}{f(re^{\iota \psi _2})}\right| +|A(re^{\iota \psi _2})|\left| \frac{f'(re^{\iota \psi _2})}{f(re^{\iota \psi _2})}\right| \\&\le r^{2\rho (f)}\{ 1+\exp {\{ (1-\epsilon )\delta (P,\psi _2)r^n\} }\} \end{aligned}$$

  which is a contradiction for sufficiently large r.

  Case 2:

  Now, suppose that there is no Borel direction of B(z) contained in $(\theta _i,\phi _{i+1})$ for any $i=1,2,\ldots , n$. In this case, $(\theta _i,\phi _{i+1})$ will be contained inside $\varOmega _{2j-1}(\Phi _{2j-1},\Phi _{2j})$, for any $j=1,2,3,\ldots , q$.

  Therefore, for $r\notin E_2\cup [0,1]$ and $\theta \in E^+\cap \varOmega _{2j-1}(\Phi _{2j-1},\Phi _{2j})$, from Eqs. (1), (6), (8), (31) and (32) we get

  $$\begin{aligned} \qquad \exp {\{ (1-\epsilon )\delta (P,\theta )r^n \} }&\le |A(re^{\iota \theta })| \\&\le \left| \frac{f''(re^{\iota \theta })}{f'(re^{\iota \theta })}\right| +|B(re^{\iota \theta })|\left| \frac{f(re^{\iota \theta })}{f'(re^{\iota \theta })}\right| \\&\le r^{2\rho (f)}+r\{ \exp {\{ -CT(r,B)\} }+|b_{2j-1}|\} \\&\le r^{2\rho (f)}(1+|b_{2j-1}|+o(1)) \end{aligned}$$

  which provides a contradiction for sufficiently large r. Thus, all non-trivial solutions f of Eq. (1) are of infinite order.

• (b) Suppose that B(z) is extremal to Denjoy’s conjecture and there exists a non-trivial solution f of Eq. (1) of finite order, then by part (i) of Lemma 1, for given $\epsilon >0$ Eq. (20) holds true for z satisfying $|z|>R$ and $\arg {z}\notin E_1$. We will discuss the following cases:

  Case 1:

  suppose that the ray $\arg {z}=\Phi$ is a Borel direction of B(z) where $\theta _i<\Phi <\phi _{i+1}$ for some $i=1,2,\ldots , n$ then the conclusion holds in similar manner as in case 1 of (a) part 1 of Theorem 2.

  Case 2:

  Suppose that $\arg {z}=\theta$ is not a Borel direction of B(z) for any $\theta \in (\theta _i,\phi _{i+1})$ for all $i=1,2,\ldots , n$, then choose $\arg {z}=\theta \in (\theta _i,\phi _{i+1})$ for some $i=1,2,\ldots , n$. Then, by Lemma 13, there exists $\sigma \in (0,\pi /4)$ such that

  $$\begin{aligned} \lim _{|z|\rightarrow \infty , z \in (\varOmega (\theta -\sigma ,\theta +\sigma )\setminus S')} \frac{\log \log |B(z)|}{\log {|z|}} =\rho (B). \end{aligned}$$

  Thus,

  $$\begin{aligned} \exp {\{ r^{\rho (B)-\epsilon }\} }\le |B(z)| \end{aligned}$$

  (35)

  for all z satisfying $|z|=r$ and $z\in (\varOmega (\theta -\sigma ,\theta +\sigma )\setminus S')\cap \left( \varOmega (\theta _i,\phi _{i+1})\setminus S\right)$, where $S=\{ z\in \mathbb {C}: \arg (z)\in E_1\}$. Now, from Eqs. (1), (7), (20) and (35) we get a contradiction for sufficiently large r. Thus, all non-trivial solutions of Eq. (1) are of infinite order.

$\square$

3.5 Proof of Part 2 of Theorem 2

Proof

If $\rho (A)\ne \rho (B)$, then from [12, Theorem 1] we have

$$\begin{aligned} \rho _2(f)= \max \{ \rho (A),\rho (B)\} . \end{aligned}$$

Therefore, we consider $\rho (A)=\rho (B) =n$ for some $n\in \mathbb {N}$.

• (a) Suppose that B(z) be entire function extremal to Yang’s inequality, then all non-trivial solutions of Eq. (1) under hypothesis are of infinite order; therefore, it follows from part (ii) of Lemma 1 that for $\epsilon >0$, there exists a set $E_2\subset [1,\infty )$ having finite logarithmic measure such that for all z satisfying $|z|=r\notin E_2\cup [0,1]$ we have

  $$\begin{aligned} \left| \frac{f^{(k)}(z)}{f^{(j)}(z)}\right| \le c \left[ T(2r,f)\right] ^{2(k-j)} \end{aligned}$$

  (36)

  where $c>0$ is a constant and $k\in \mathbb {N}$. Now, we have two cases to deal with:

  Case 1:

  let us suppose that there is a Borel direction $\Phi$ of B(z) such that $\theta _i<\Phi <\phi _{i+1}$ for any $i=1,2,3, \ldots ,n$. Thus, for $r\notin E_2\cup [0,1]$ from Eqs. (1), (7), (34) and (36) we have

  $$\begin{aligned} \exp {\{ r^{n-\epsilon }\} }&\le |B(re^{\iota \psi _2})|\le \left| \frac{f''(re^{\iota \psi _2})}{f(re^{\iota \psi _2})}\right| +|A(re^{\iota \psi _2})|\left| \frac{f'(re^{\iota \psi _2})}{f(re^{\iota \psi _2})}\right| \\&\le c \left[ T(2r,f)\right] ^4+ \exp {\{ (1-\epsilon )\delta (P,\psi _2)r^n\} }c \left[ T(2r,f)\right] ^2\\&\le c\left[ T(2r,f)\right] ^4(1+o(1)) \end{aligned}$$

  for sufficiently large $r\notin E_2\cup [0,1]$. Thus,

  $$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log {\log {T(r,f)}}}{\log {r}}\ge n. \end{aligned}$$

  (37)

  Then, it can be seen easily from Eq. (37) and Lemma 6 that

  $$\begin{aligned} \rho _2(f)=n \end{aligned}$$

  for all non-trivial solutions f of Eq. (1).

  Case 2:

  Now, suppose that there is no Borel direction of B(z) contained in $(\theta _i,\phi _{i+1})$ for any $i=1,2,\ldots , n$. Then, from Eqs. (1), (6), (8), (32) and (36), we get

  $$\begin{aligned} \exp {\{ (1-\epsilon )\delta (P,\theta )r^n\} }&\le |A(re^{\iota \theta })| \\&\le \left| \frac{f''(re^{\iota \theta })}{f'(re^{\iota \theta })}\right| +|B(re^{\iota \theta })|\left| \frac{f(re^{\iota \theta })}{f'(re^{\iota \theta })}\right| \\&\le c \left[ T(2r,f)\right] ^2+r \{ |b_{2j-1}|+\exp {\{ -CT(r,B)\} }\} \\&\le \mathrm{d}r \left[ T(2r,f)\right] ^2\left( 1+|b_{2j-1}|+o(1)\right) \end{aligned}$$

  for sufficiently large $r\notin E_2\cup [0,1]$, $\theta \in E^+\cap \varOmega _{2j-1}(\Phi _{2j-1},\Phi _{2j})$ and for $d>0$ is a constant. Thus,

  $$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log {\log {T(r,f)}}}{\log {r}}\ge n. \end{aligned}$$

  (38)

  It follows from Eq. (38) and Lemma 6 that

  $$\begin{aligned} \rho _2(f)=n \end{aligned}$$

  for all non-trivial solution f of Eq. (1).

• (b) Suppose B(z) is an entire function extremal to Denjoy’s conjecture, we again discuss two cases:

  Case 1:

  suppose that the ray $\arg {z}=\Phi$ is a Borel direction of B(z) where $\theta _i<\Phi <\phi _{i+1}$ for some $i=1,2,\ldots , n$, then the conclusion holds in similar manner as in case 1 of (a) of part 1 of Theorem 2.

  Case 2:

  Suppose that $\arg {z}=\theta$ is not a Borel direction of B(z) for any $\theta \in (\theta _i,\phi _{i+1})$ for all $i=1,2,\ldots , n$, choose $\arg {z}=\theta \in (\theta _i,\phi _{i+1})$ for some $i=1,2,\ldots , n$. Then, from Eqs. (1), (4), (7) and (35), we obtain

  $$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log \log T(r,f)}{\log {r}}\ge n. \end{aligned}$$

  (39)

  From Lemma 6 and Eq. (39), we get the desired result.

$\square$

4 Extension to Higher Order

This section involves linear differential equation of the form:

$$\begin{aligned} f^{(k)}+A_{k-1}(z)f^{(k-1)}+A_{k-2}f^{(k-2)}+\cdots +A_{0}f=0 \end{aligned}$$

(40)

where $A_{k-1}(z), A_{k-2}(z),\ldots , A_0(z)$ are entire functions; therefore, all solutions of Eq. (40) are entire functions [14]. Also, all solutions of Eq. (40) are of finite order if and only if all coefficients $A_{k-1}(z), A_{k-2}(z),\ldots , A_0(z)$ are polynomials [14, Theorem 4.1]. Thus, if any of the coefficient of Eq. (40) is a transcendental entire function, then there will exist a non-trivial solution of infinite order. In this section, we will discuss conditions on coefficients $A_{k-1}(z), A_{k-2}(z),\ldots , A_0(z)$ so that all non-trivial solutions of Eq. (40) are of infinite order. For this purpose, we will extend our previous results of [11] in the following manner:

Theorem 3

Suppose that there exists an integer $i\in \{ 1,2, \ldots , k-1\}$ such that $\lambda (A_i)<\rho (A_i)=n, n\in \mathbb {N}$, $A_0(z)$ is a transcendental entire function satisfying $\mu (A_0)\ne \rho (A_i)$ and $\rho (A_j)<\mu (A_0)$ for all $j=1,2,\ldots , k-1$, $j\ne i$, then all non-trivial solutions of Eq. (40) are of infinite order. Moreover, for these solutions f, we have

$$\begin{aligned} \rho _2(f)\ge \mu (A_0) . \end{aligned}$$

Corollary 2

If $A_0(z)$ is a transcendental entire function in the hypothesis of Theorem 3 such that $\mu (A_0)\ne \mu (A_i)$ and $\rho (A_j)<\mu (A_0)$ for all $j=1,2,\ldots , k-1$, $j\ne i$, then the conclusion of Theorem 3 holds true.

Theorem 4

Suppose $A_1(z)=v(z)e^{P(z)}$ is an entire function with $\lambda (A_1) < \rho (A_1)$, where P(z) is non-constant polynomial and v(z) is an entire function with $\rho (v)<\rho (A_1)$. If $A_0(z)$ is an entire function extremal to Yang’s inequality such that no Borel direction of $A_0(z)$ coincides with any of the critical rays of $e^{P(z)}$ and $\rho (A_j)<\rho (A_0)$, where $j=2,\ldots , k-1$, then all non-trivial solutions f of Eq. (40) satisfy

$$\begin{aligned} \rho (f)=\infty \quad \text{ and } \quad \rho _2(f)\ge \rho (A_0) . \end{aligned}$$

Theorem 5

Suppose that there exists an integer $i\in \{ 1,2,\ldots , k-1\}$ such that $\lambda (A_i)<\rho (A_i)$, $A_0(z)$ is an entire function extremal to Denjoy’s conjecture and $\rho (A_j)<\rho (A_0), j=1,2,\ldots , k-1, j\ne i$, then all non-trivial solutions f of Eq. (40) satisfy

$$\begin{aligned} \rho (f)=\infty \quad \text{ and } \quad \rho _2(f)\ge \rho (A_0) . \end{aligned}$$

5 Proof of Theorem 3

Proof

Assume that there exists a non-trivial solution f of Eq. (40) with finite order. Then, by part (ii) of Lemma 1, for given $\epsilon >0$ there exists a set $E_2\subset (1,\infty )$ that has finite logarithmic measure such that

$$\begin{aligned} \left| \frac{f^{(m)}(z)}{f(z)}\right| \le |z|^{k(\rho (f)+\epsilon )}, \quad m=1,2,3,\ldots , k \end{aligned}$$

(41)

for all z satisfying $|z|\not \in E_2\cup [0,1]$.

Case 1:

Suppose $\rho (A_i)<\mu (A_0)$. Then, from Eqs. (40) and (41), we get

$$\begin{aligned} \left| A_0(z)\right|&\le \left| \frac{f^{(k)}(z)}{f(z)}\right| +|A_{k-1}(z)|\left| \frac{f^{(k-1)}(z)}{f(z)}\right| +\cdots +|A_1(z)|\left| \frac{f'(z)}{f(z)}\right| \\&\le |z|^{k(\rho (f)+\epsilon )}\left[ 1+|A_{k-1}(z)|+\cdots +|A_1(z)|\right] \end{aligned}$$

for all z satisfying $|z|\not \in E_2\cup [0,1]$ which implies that

$$\begin{aligned} T(r,A_0)\le k\left( \rho (f)+\epsilon \right) \log {r}+ (k-1)T(r,A_m)+O(1) \end{aligned}$$

where $T(r,A_m)=\max \{ T(r,A_p): p=1,2,\ldots , k-1 \}$ and $|z|=r\notin E_2\cup [0,1]$. This gives that $\mu (A_0)\le \mu (A_m), m=1,2, \ldots , k-1$ which is a contradiction. Thus, all non-trivial solutions of Eq. (40) are of infinite order. Suppose that f is a non-trivial solution of Eq. (40) then by part (ii) of Lemma 1, for $\epsilon >0$ there exists a set $E_2\subset (1,\infty )$ that has finite logarithmic measure and there exists a constant $c>0$ such that for all z satisfying $|z|=r\notin E_2\cup [0,1]$ we have

$$\begin{aligned} \left| \frac{f^{(m)}(z)}{f(z)}\right| \le c \left( T(2 r,f)\right) ^{2k}\quad m=1,2,\ldots , k \end{aligned}$$

(42)

Choose $\max \{ \rho (A_p): p=1,2,\ldots k-1\}<\eta < \mu (A_0)$, then from Eqs. (40), (42) and Remark 2 we get

$$\begin{aligned} \exp {\left( r^{\mu (A_0)-\epsilon }\right) }&\le M(r,A_0)\\&\le cT(2r,f)^{2k}\left( 1+M(r,A_{k-1}) +\cdots +M(r,A_1) \right) \\&\le cT(2r,f)^{2k}[1+(k-1)\exp {(r^{\eta })}] \end{aligned}$$

for all z satisfying $|z|=r\notin E_2$. This implies that

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log \log T(r,f)}{\log r}\ge \mu (A_0). \end{aligned}$$

Case 2:

When $\mu (A_0)<\rho (A_i)=n$, where $n\in \mathbb {N}$ and there is a non-trivial solution f of Eq. (40) of finite order then by part (i) of Lemma 1, for given $\epsilon >0$ there exists a set $E_1 \subset [0,2\pi )$ that has linear measure zero such that if $\psi _0 \in [0,2\pi )\setminus E_1,$ then there is a constant $R_0=R(\psi _0)>0$ so that for all z satisfying $\arg z =\psi _0$ and $|z| \ge R_0$ we have

$$\begin{aligned} \left| \frac{f^{(m)}(z)}{f(z)}\right| \le |z|^{k\rho (f)} \qquad m=1,2,\ldots ,k. \end{aligned}$$

(43)

Also $\rho (A_j)<\mu (A_0)$ for $j=1,2,\ldots ,k-1, j\ne i$, then we can choose $\eta >0$ such that

$$\begin{aligned} \max \{ \rho (A_j), j=1,2,\ldots .k-1, j\ne i\}<\eta <\mu (A_0). \end{aligned}$$

From above, we have that

$$\begin{aligned} |A_j(z)|\le \exp \{r^{\eta }\}, \quad j=1,2,\ldots , k-1, j\ne i. \end{aligned}$$

(44)

We have following cases to discuss:

Subcase 1:

when $0<\mu (A_0)<1/2$, then Lemma 8, Eqs. (7), (40), (43) and (44) give

$$\begin{aligned} \exp {\left( r^{\mu (A_0)-\epsilon }\right) }&\le |z|^{k\rho (f)}\left[ 1+\exp {\left( (1-\epsilon )\delta (P, \theta )r^n\right) }+(k-2)\exp {r^{\eta }}\right] \end{aligned}$$

for all z satisfying $|z|=r>R$ and $\arg {z}\in E^-\setminus (E_1\cup E)$. This gives a contradiction for sufficiently large r.

Subcase 2:

Assume that $\mu (A_0)\ge 1/2$, then by Lemma 9, Eqs. (7), (40), (43) and (44), we get a contradiction.

Subcase 3:

Suppose that $\mu (A_0)=0$, then using Lemma 7, Eqs. (7), (40), (43) and (44), we again get a contradiction. Therefore, all non-trivial solutions of Eq. (40) are of infinite order.

If $\mu (A_0)=0$, then $\rho _2(f)\ge 0$ for all non-trivial solutions f of Eq. (40). Therefore, we suppose that $\mu (A_0)>0$ then using Lemma 8 (or Lemma 9 ), Eqs. (7), (40), (42) and (44) we get

$$\begin{aligned} \rho _2(f)\ge \mu (A_0) \end{aligned}$$

for all non-trivial solutions f of Eq. (40).$\square$

6 Proof of Theorem 4

Proof

When $\rho (A_0)\ne \rho (A_1)$, then the result follows from [11, Theorem 5]. Thus, we need to consider $\rho (A_0)=\rho (A_1)=n$ where $n\in \mathbb {N}$. Let us suppose that there exists a non-trivial solution f of Eq. (40) of finite order. Then, from part (ii) of Lemma 1, for $\epsilon >0$ there exists a set $E_2 \subset (1,\infty )$ with finite logarithmic measure such that

$$\begin{aligned} \left| \frac{f^{(m)}(z)}{f^{(p)}(z)}\right| \le |z|^{(m-p)\rho (f)}, \quad m, p=0, 1,2,\ldots , k, p<m \end{aligned}$$

(45)

for all z satisfying $|z|=r\notin E_2\cup [0,1]$.

Since $\rho (A_j)<\rho (A_0)$ for $j=2,3,\ldots , k-1$, then we choose $\eta >0$ such that

$$\begin{aligned} \max \{ \rho (A_j):j=2,3, \ldots ,k-1\}<\eta <\rho (A_0) \end{aligned}$$

so that

$$\begin{aligned} |A_j(z)|\le \exp {r^{\eta }} \end{aligned}$$

(46)

for $j=2,3,\ldots , k-1.$ As done earlier in Theorem 2, we have following two cases to consider:

Case 1:

if there exists a Borel direction $\Phi$ of $A_0(z)$ such that $\theta _i<\Phi <\phi _{i+1}$ for $i=1,2,\ldots ,n$, then from Eqs. (7), (34), (40), (45) and (46) we have

$$\begin{aligned} \exp {\left( r^{n-\epsilon }\right) }&\le |A_0(z)|\le \left| \frac{f^{(k)}(z)}{f(z)}\right| +|A_{(k-1)}(z)|\left| \frac{f^{(k-1)}(z)}{f(z)}\right| +\\&\cdots +|A_1(z)|\left| \frac{f'(z)}{f(z)}\right| \\&\le |z|^{k\rho (f)}[1+(k-2)\exp {(r^{\eta })}+\exp {(1-\epsilon )\delta (P,\psi _2)}] \end{aligned}$$

for all z satisfying $|z|=r\notin E_2\cup [0,1]$ and $\arg {z}=\psi _2$. This will lead us to a contradiction for large values of r. Thus, all non-trivial solutions of Eq. (40) are of infinite order. From Eqs. (7), (34), (40), (42) and (46), we have $\rho _2(f)\ge \rho (A_0)$ for all non-trivial solutions f of Eq. (40).

Case 2:

If there does not exists any Borel direction of $A_0(z)$ contained in $(\theta _i,\phi _{i+1})$ for $i=1,2,\ldots , n$, then from Eqs. (6), (8), (32), (40), (45) and (46) we have

$$\begin{aligned} \exp {((1-\epsilon )\delta (P,\theta )r^n)}&\le |A_1(z)|\le \left| \frac{f^{(k)}(z)}{f'(z)}\right| +|A_{(k-1)}(z)|\left| \frac{f^{(k-1)}(z)}{f'(z)}\right| +\\&\cdots +|A_2(z)|\left| \frac{f''(z)}{f(z)}\right| +|A_0(z)|\left| \frac{f(z)}{f'(z)}\right| \\&\le |z|^{k\rho (f)}[1+(k-2)\exp (r^{\eta })\\&+r(\exp {\left( -CT(r,A_0)\right) }+|a_{2j-1}|)] \end{aligned}$$

for all $|z|=r\notin E_2\cup [0,1]$ and $\arg {z}=\theta \in E^+\cap \varOmega _{2j-1}(\Phi _{2j-1}, \phi _{2j})$, where $a_i, i=1,2,\ldots q/2$ are deficient values of $A_0(z)$. This gives a contradiction for sufficiently large r. Thus, all non-trivial solutions of Eq. (40) are of infinite order.

From Eqs. (6), (8), (32), (40), (42) and (46), we have $\rho _2(f)\ge \rho (A_0)$ for all non-trivial solutions f of Eq. (40). $\square$

7 Proof of Theorem 5

Proof

If $\rho (A_i)\ne \rho (A_0)$, then result is true from [11, Theorem 5]. Assume that $\rho (A_i)=\rho (A_0)=n, n\in \mathbb {N}$ and there exists a non-trivial solution f of Eq. (40) of finite order. Then, we have following two cases to discuss:

Case 1:

when the ray $\arg {z}=\Phi$ is a Borel direction of $A_0(z)$ where $\Phi \in (\theta _i,\phi _{i+1})$ for some $i=1,2,\ldots , n$, choose $\psi _1<\psi _2$ such that $\theta _i<\psi _1<\phi<\psi _2<\phi _{i+1}$ and $\psi _2-\psi _1<\pi /\rho (A_i)=\pi /\rho (A_0)$. Then, by Lemma 12 we can have

$$\begin{aligned} \limsup _{r\rightarrow \infty }\frac{\log {\log {|A_0(re^{\iota \psi _2})|}}}{\log {r}}=\rho (A_0) \end{aligned}$$

(47)

Thus, from Eqs. (7), (40), (45), (46) and (47) we get contradiction for sufficiently large r. As done in case 1 of Theorem 4, we get $\rho _2(f)\ge \rho (A_0)$ for all non-trivial solutions f of Eq. (40).

Case 2:

Suppose that $\arg {z}=\theta$ is not a Borel direction of $A_0(z)$ for any $\theta \in (\theta _i,\phi _{i+1})$ for all $i=1,2,\ldots , n$, then choose $\arg {z}=\theta \in (\theta _i,\phi _{i+1})$ for some $i=1,2,\ldots , n$. Then, Lemma 13, Eqs. (5), (7), (40) and (44) lead us to a contradiction for sufficiently large r. Thus, all non-trivial solutions of Eq. (40) are of infinite order.

Also, Lemma 13, Eqs. (4), (7), (40) and (44) give that $\rho _2(f)\ge \rho (A_0)$ for all non-trivial solutions f of Eq. (40).