Classes of harmonic functions associated with Ruscheweyh derivatives

Abstract

In the paper we introduce the classes of functions defined by generalized Ruscheweyh derivatives and we show that they can be presented as dual sets. Moreover, by using extreme points theory, we obtain estimations of classical convex functionals on the defined classes of functions. Some applications of the main results are also considered.

1 Introduction

A real-valued function u is said to be harmonic in a domain $D\subset \mathbb {C}$ if it has continuous second order partial derivatives in D,  which satisfy the Laplace equation

$$\begin{aligned} \Delta u:=\frac{\partial ^{2}u}{\partial x^{2}}+\frac{\partial ^{2}u}{ \partial y^{2}}=0. \end{aligned}$$

We say that a complex-valued continuous function $f:D\rightarrow \mathbb {C}$ is harmonic in D if both functions $u:=\mathrm {Re}f$ and $v:=\mathrm {Im} f$ are real-valued harmonic functions in D. We note that every complex-valued function f harmonic in D with $0\in D,$ can be uniquely represented as

$$\begin{aligned} f=h+\overline{g}, \end{aligned}$$

(1)

where h, g are analytic functions in D with $g(0)=0$. Then we call h the analytic part and g the co-analytic part of f (see [4]). It is easy to verify, that the Jacobian of f is given by

$$\begin{aligned} J_{f}\left( z\right) =\left| h^{\prime }\left( z\right) \right| ^{2}-\left| g^{\prime }\left( z\right) \right| ^{2}\ \ \ \left( z\in D\right) . \end{aligned}$$

The mapping f is locally univalent if $J_{f}\left( z\right) \ne 0$ in D. A result of Lewy [15] shows that the converse is true for harmonic mappings. Therefore, f is locally univalent and sense-preserving if and only if

$$\begin{aligned} \left| h^{\prime }\left( z\right) \right| >\left| g^{\prime }\left( z\right) \right| \ \ \ \left( z\in D\right) . \end{aligned}$$

(2)

Let $\mathcal {H}$ denote the class of harmonic functions in the unit disc ${ \mathbb {U}}.$ Any function $f\in \mathcal {H}$ can be written in the form

$$\begin{aligned} f(z)=\sum \limits _{n=0}^{\infty }a_{n}z^{n}+\sum \limits _{n=1}^{\infty } \overline{b_{n}z^{n}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

(3)

Let $\mathbb {N}_{l}:=\left\{ l,l+1, \ldots \right\} ,$$\mathbb {N}:=\mathbb {N} _{1},\ k\in \mathbb {N}_{2},$ and let $\mathcal {H}\left( k\right)$ denote the class of function with missing coefficients i.e. the functions $f\in \mathcal {H}$ of the form

$$\begin{aligned} f(z)=z+\sum \limits _{n=k}^{\infty }\left( a_{n}z^{n}+\overline{b_{n}z^{n}} \right) \ \ \ \left( z\in \mathbb {U}\right) , \end{aligned}$$

(4)

which are univalent and sense-preserving in ${\mathbb {U}}.$

We say that a function $f\in \mathcal {H}\left( 2\right)$ is harmonic starlike in ${\mathbb {U}}\left( r\right)$ if

$$\begin{aligned} \frac{\partial }{\partial t}\left( \arg f\left( re^{it}\right) \right) >0\ \left( 0\le t\le 2\pi \right) \end{aligned}$$

i.e. f maps the circle $\partial {\mathbb {U}}\left( r\right)$ onto a closed curve that is starlike with respect to the origin. It is easy to verify, that the above condition is equivalent to the following

$$\begin{aligned} \mathrm {Re}\,\frac{D_{\mathcal {H}}f\left( z\right) }{f\left( z\right) }>0\ \ \ \left( \left| z\right| =r\right) , \end{aligned}$$

where

$$\begin{aligned} D_{\mathcal {H}}f\left( z\right) :=zh^{\prime }\left( z\right) -\overline{ zg^{\prime }\left( z\right) }\ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Ruscheweyh [20] introduced an operator $\mathcal {D}^{m}:\,\mathcal {A} \rightarrow \mathcal {A},$ defined by

$$\begin{aligned} \mathcal {D}^{m}f(z)=\frac{z\left( z^{m-1}f(z)\right) ^{(m)}}{m!}\quad ( m\in \mathbf {\mathbb {N}}_{0},\ z\in \mathbb {U}). \end{aligned}$$

(5)

The Ruscheweyh derivative $\mathcal {D}^{m}$ was extended in [17] (see also [6, 8, 10, 23]) on the class of harmonic functions. Let $D_{ \mathcal {H}}^{m}:\mathcal {H}\rightarrow \mathcal {H}$ denote the linear operator defined for a function $f=h+\overline{g}\in \mathcal {H}$ by

$$\begin{aligned} D_{\mathcal {H}}^{m}f:=\mathcal {D}^{m}h+( -\,1) ^{m}\overline{ \mathcal {D}^{m}g}. \end{aligned}$$

We say that a function $f\in \mathcal {H}$ is subordinate to a function $F\in \mathcal {H}$, and write $f(z)\prec F(z)$ (or simply $f\prec F$) if there exists a complex-valued function $\omega \$which maps $\mathbb {U}$ into oneself with $\omega (0)=0$, such that $f(z)=F(\omega (z))\ \ \ \left( z\in \mathbb {U}\right) .$

Let A, B be complex parameters, $A\ne B{.}$ We denote by $\mathcal {S}_{ \mathcal {H}}^{m}(k;A,B)\,$the class of functions $f\in \mathcal {H}\left( k\right)$ such that

$$\begin{aligned} \frac{D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) }{D_{ \mathcal {H}}^{m}f\left( z\right) }\prec \frac{1+Az}{1+Bz}. \end{aligned}$$

(6)

Also, by $\mathcal {R}_{\mathcal {H}}^{m}(k;A,B)\,$we denote the class of functions $f\in \mathcal {H}\left( k\right)$ such that

$$\begin{aligned} \frac{D_{\mathcal {H}}^{m}f\left( z\right) }{z}\prec \frac{1+Az}{1+Bz}. \end{aligned}$$

The classes $\mathcal {S}_{\mathcal {H}}^{m}(k;A,B)$ and $\mathcal {R}_{ \mathcal {H}}^{m}(k;A,B)$ with restrictions ${-B\le A<B\le 1}$, $k=2,$ were investigated in [8]. In particular, the class

$$\begin{aligned} \mathcal {S}_{\mathcal {H}}^{m}(\alpha ):=\mathcal {S}_{\mathcal {H} }^{m}(2\alpha -1,1)\ \ \ \left( 0\le \alpha <1\right) \end{aligned}$$

is related to the class of Sălăgean-type harmonic functions studied by Yalçin [22]. The classes

$$\begin{aligned} \mathcal {R}_{\mathcal {H}}(k;A,B):=\mathcal {R}_{\mathcal {H}}^{1}(k;A,B),\ \mathcal {S}_{\mathcal {H}}(k;A,B):=\mathcal {S}_{\mathcal {H}}^{0}(k;A,B),\ \mathcal {K}_{\mathcal {H}}(k;A,B):=\mathcal {S}_{\mathcal {H}}^{1}(k;A,B) \end{aligned}$$

are defined in [6] (see also [7]).

The object of the present paper is to show that the defined classes of functions can be presented as dual sets. Also, by using extreme points theory, we obtain estimations of classical convex functionals on the defined classes of functions with correlated coefficients. Some applications of the main results are also considered.

2 Dual sets

For functions $f_{1},f_{2}\in \mathcal {H}$ of the form of the form

$$\begin{aligned} f_{l}(z)=\sum \limits _{k=0}^{\infty }\left( a_{l,k}z^{k}+\overline{ b_{l,k}z^{k}}\right) \ \ \ \left( z\in \mathbb {U},l\in \mathbb {N}\right) \end{aligned}$$

(7)

we define the Hadamard product or convolution of $f_{1}$ and $f_{2}$ by

$$\begin{aligned} \left( f_{1}*f_{2}\right) \left( z\right) =\sum \limits _{k=0}^{\infty }\left( a_{1,k}a_{2,k}z^{k}+\overline{b_{1,k}b_{2,k}z^{k}}\right) \ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Let $\mathcal {V}\subset \mathcal {H}$, $\mathbb {U}_{0}:=\mathbb {U}\smallsetminus \left\{ 0\right\} .$ Motivated by Ruscheweyh [19] we define the dual set of $\mathcal {V}$ by

$$\begin{aligned} \mathcal {V}^{*}:=\left\{ f\in \mathcal {H}\left( k\right) :\bigwedge \limits _{g\in \mathcal {V}}\left( f*g\right) \left( z\right) \ne 0{\ \ \ \left( z\in \mathbb {U}_{0}\right) }\right\} . \end{aligned}$$

Theorem 1

$$\begin{aligned} \mathcal {S}_{\mathcal {H}}^{m}(k;A,B)=\left\{ D_{\mathcal {H}}^{m}\left( \psi _{\xi }\right) :\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right):= & {} z\frac{1+B\xi -\left( 1+A\xi \right) \left( 1-z\right) }{\left( 1-z\right) ^{2}} \\&-\overline{z}\frac{1+B\xi +\left( 1+A\xi \right) \left( 1-\overline{z} \right) }{\left( 1-\overline{z}\right) ^{2}}\ \ \ \left( z\in \mathbb {U} \right) . \end{aligned}$$

Proof

Let $f\in \mathcal {H}\left( k\right)$ be of the form (1). Then $f\in \mathcal {S}_{\mathcal {H}}^{m}(k;A,B)$ if and only if it satisfies (6) or equivalently

$$\begin{aligned} \frac{D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) }{D_{ \mathcal {H}}^{m}f\left( z\right) }\ne \frac{1+A\xi }{1+B\xi }\ \ \ \left( z\in \mathbb {U}_{0},~\left| \xi \right| =1\right) . \end{aligned}$$

Since

$$\begin{aligned} D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}h\right) \left( z\right) =D_{ \mathcal {H}}^{m}h\left( z\right) *z/(1-z)^{2},\ D_{\mathcal {H} }^{m}h\left( z\right) =D_{\mathcal {H}}^{m}h\left( z\right) *\frac{z}{1-z} , \end{aligned}$$

the above inequality yields

$$\begin{aligned}&\left( 1+B\xi \right) D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -\left( 1+A\xi \right) D_{\mathcal {H}}^{m}f\left( z\right) \\&\quad =\left( 1+B\xi \right) D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}h\right) \left( z\right) -\left( 1+A\xi \right) D_{\mathcal {H}}^{m}h\left( z\right) \\&\qquad -\left( -1\right) ^{m}\left[ \left( 1+B\xi \right) \overline{D_{\mathcal {H} }\left( D_{\mathcal {H}}^{m}g\right) \left( z\right) }+\left( 1+A\xi \right) \overline{D_{\mathcal {H}}^{m}g\left( z\right) }\right] \\&\quad =D_{\mathcal {H}}^{m}h\left( z\right) *\left( \frac{\left( 1+B\xi \right) z}{\left( 1-z\right) ^{2}}-\frac{\left( 1+A\xi \right) z}{1-z}\right) \\&\qquad -\left( -1\right) ^{m}\overline{D_{\mathcal {H}}^{m}g\left( z\right) }*\left( \frac{\left( 1+B\xi \right) \overline{z}}{\left( 1-\overline{z} \right) ^{2}}+\frac{\left( 1+A\xi \right) \overline{z}}{1-\overline{z}} \right) \\&\quad =f\left( z\right) *D_{\mathcal {H}}^{m}\psi _{\xi }\left( z\right) \ne 0\ \ \ \left( z\in \mathbb {U}_{0},~\left| \xi \right| =1\right) . \end{aligned}$$

Thus, $f\in \mathcal {S}_{\mathcal {H}}^{m}(k;A,B)$ if and only if $f\left( z\right) *D_{\mathcal {H}}^{m}\psi _{\xi }\left( z\right) \ne 0$ for $z\in \mathbb {U}_{0},~\left| \xi \right| =1$i.e.$\mathcal {S}_{\mathcal {H}}^{m}(k;A,B)=\left\{ D_{\mathcal {H}}^{m}\left( \psi _{\xi }\right) :\ \left| \xi \right| =1\right\} ^{*}$. $\square$

Similarly as Theorem 1 we prove the following theorem.

Theorem 2

$$\begin{aligned} \mathcal {R}_{\mathcal {H}}^{m}(k;A,B)=\left\{ \delta _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \delta _{\xi }\left( z\right) :=z\frac{1+B\xi -\left( 1+A\xi \right) \left( 1-z\right) ^{m+1}}{\left( 1-z\right) ^{m+1}}+\left( -1\right) ^{m}\overline{z }\frac{1+B\xi }{\left( 1-\overline{z}\right) ^{m+1}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

In particular, by Theorems 1 and 2 we obtain the following results.

Theorem 3

$$\begin{aligned} \mathcal {S}_{\mathcal {H}}^{m}(\alpha )=\left\{ D_{\mathcal {H}}^{m}\left( \psi _{\xi }\right) :\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right):= & {} z\frac{1+\xi -\left( 1-\xi +2\alpha \xi \right) \left( 1-z\right) }{\left( 1-z\right) ^{2}} \\&-\,\overline{z}\frac{1+\xi +\left( 1-\xi +2\alpha \xi \right) \left( 1- \overline{z}\right) }{\left( 1-\overline{z}\right) ^{2}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Theorem 4

$$\begin{aligned} \mathcal {S}_{\mathcal {H}}(k;A,B)=\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right) :=z\frac{\left( B-A\right) \xi +\left( 1+A\xi \right) z}{\left( 1-z\right) ^{2}}-\overline{z}\frac{2+\left( A+B\right) \xi -\left( 1+A\xi \right) \overline{z}}{\left( 1-\overline{z}\right) ^{2}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Theorem 5

$$\begin{aligned} \mathcal {K}_{\mathcal {H}}(k;A,B)=\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right) :=z\frac{\left( B-A\right) \xi +\left( 2+A\xi +B\xi \right) z}{\left( 1-z\right) ^{3}}+\overline{z}\frac{2+\left( A+B\right) \xi +\left( B-A\right) \xi \overline{z}}{\left( 1-\overline{z} \right) ^{3}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Theorem 6

$$\begin{aligned} \mathcal {R}_{\mathcal {H}}(k;A,B)=\left\{ \delta _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \delta _{\xi }\left( z\right) :=z\frac{\left( 1+B\xi \right) -\left( 1+A\xi \right) \left( 1-z\right) ^{2}}{\left( 1-z\right) ^{2}}-\frac{\left( 1+B\xi \right) \overline{z}}{\left( 1-\overline{z}\right) ^{2}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

3 Correlated coefficients

Let us consider the function $\varphi \in \mathcal {H}$ of the form

$$\begin{aligned} \varphi =u+\overline{v},\ \ u(z)=\sum \limits _{n=0}^{\infty }u_{n}z^{n},\ \ v(z)=\sum \limits _{n=1}^{\infty }v_{n}z^{n}\ \ \ \left( z\in \mathbb {U} \right) . \end{aligned}$$

(8)

We say that a function $f\in \mathcal {H}$ of the form (4) has coefficients correlated with the function $\varphi ,$ if

$$\begin{aligned} u_{n}a_{n}=-\left| u_{n}\right| \left| a_{n}\right| ,\ v_{n}b_{n}=\left| v_{n}\right| \left| b_{n}\right| \ \ \left( n\in \mathbb {N}_{k}\right) . \end{aligned}$$

(9)

In particular, if there exists a real number $\eta$ such that

$$\begin{aligned} \varphi \left( z\right) =\frac{z}{1-{e^{i\eta }}z}+\frac{\overline{z}}{1-{ e^{i\eta }}\overline{z}}=\sum \limits _{n=1}^{\infty }{e^{i\left( n-1\right) \eta }}\left( z^{n}+\overline{z}^{n}\right) \ \ \ \left( z\in \mathbb {U} \right) , \end{aligned}$$

then we obtain functions with varying coefficients defined by Jahangiri and Silverman [11] (see also [7]). Moreover, if we take

$$\begin{aligned} \varphi \left( z\right) =2{\mathfrak {R}}\frac{z}{1-z}=\sum \limits _{n=1}^{\infty }\left( z^{n}+\overline{z}^{n}\right) \ {\ }\ \left( z\in \mathbb {U}\right) , \end{aligned}$$

then we obtain functions with negative coefficients introduced by Silverman [21]. These functions were intensively investigated by many authors (for example, see [5,6,7,8,9, 11, 13, 25]).

Let $\mathcal {T}^{m}\left( k,\eta \right)$ denote the class of functions $f\in \mathcal {H}\left( k\right)$ with coefficients correlated with respect to the function

$$\begin{aligned} \varphi \left( z\right) =\frac{z}{(1-{e^{i\eta }}z)^{m+1}}{+}\frac{\left( -1\right) ^{m}\overline{z}}{(1-{e^{i\eta }}\overline{z})^{m+1}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

(10)

Moreover, let us define

$$\begin{aligned} \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B):=\mathcal {T}^{m}\left( k,\eta \right) \cap \mathcal {S}_{\mathcal {H}}^{m}(k;A,B),\ \mathcal {R}_{\mathcal {T} }^{m}(k,\eta ;A,B):=\mathcal {T}^{m}\left( k,\eta \right) \cap \mathcal {R}_{ \mathcal {H}}^{m}(k;A,B), \end{aligned}$$

where $\eta ;A,B$ are real parameters with $B>\max \{0,A\}.$

Let $f\in \mathcal {H}\left( k\right)$ be of the form (4). Thus, by (5) we have

$$\begin{aligned} D_{\mathcal {H}}^{m}f\left( z\right) =z+\sum \limits _{n=k}^{\infty }\lambda _{n}a_{n}z^{n}+\left( -1\right) ^{m}\sum \limits _{n=k}^{\infty }\lambda _{n} \overline{b_{n}}\overline{z}^{n}\ \ \ \left( z\in \mathbb {U}\right) , \end{aligned}$$

where

$$\begin{aligned} \lambda _{n}:=\frac{\left( m+1\right) \cdot \cdots \cdot (m+n-1)}{(n-1)!}\ \ \ \left( n\in \mathbf {\mathbb {N}}_{k}\right) . \end{aligned}$$

(11)

Theorem 7

If a function $f\in \mathcal {H}\left( k\right)$ of the form (4) satisfies the condition

$$\begin{aligned} \sum \limits _{n=k}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) \le B-A, \end{aligned}$$

(12)

where

$$\begin{aligned} \alpha _{n}=\lambda _{n}\left\{ n\left( 1+B\right) -\left( 1+A\right) \right\} ,\ \ \beta _{n}=\lambda _{n}\left\{ n\left( 1+B\right) +\left( 1+A\right) \right\} , \end{aligned}$$

(13)

then $f\in \mathcal {S}_{\mathcal {H}}^{m}(k;A,B)$.

Proof

It is clear that the theorem is true for the function $f\left( z\right) \equiv z.$ Let $f\in \mathcal {H}\left( k\right)$ be a function of the form (4) and let there exist $n\in \mathbb {N}_{k}$ such that $a_{n}\ne 0$ or $b_{n}\ne 0.$ Since $\lambda _{n}\ge \lambda _{k}\ge 1,$we have

$$\begin{aligned} \frac{\alpha _{n}}{B-A}\ge n,\ \frac{\beta _{n}}{B-A}\ge n,\ \ \ n\in \mathbb {N} _{k}, \end{aligned}$$

(14)

Thus, by (12) we get

$$\begin{aligned} \sum \limits _{n=k}^{\infty }\left( n\left| a_{n}\right| +n\left| b_{n}\right| \right) \le 1 \end{aligned}$$

(15)

and

$$\begin{aligned} \left| h^{\prime }\left( z\right) \right| -\left| g^{\prime }\left( z\right) \right|&\ge 1-\sum \limits _{n=k}^{\infty }n\left| a_{n}\right| \left| z\right| ^{n}-\sum \limits _{n=k}^{\infty }n\left| b_{n}\right| \left| z\right| ^{n}\ge 1-\left| z\right| \sum \limits _{n=k}^{\infty }\left( n\left| a_{n}\right| +n\left| b_{n}\right| \right) \\&\ge 1-\frac{\left| z\right| }{B-A}\sum \limits _{n=k}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) \ge 1-\left| z\right| >0\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Therefore, by (2) the function f is locally univalent and sense-preserving in $\mathbb {U}$. Moreover, if $z_{1},z_{2}\in \mathbb {U},$$z_{1}\ne z_{2},$ then.

$$\begin{aligned} \left| \frac{z_{1}^{n}-z_{2}^{n}}{z_{1}-z_{2}}\right| =\left| \sum \limits _{l=1}^{n}z_{1}^{l-1}z_{2}^{n-l}\right| \le \sum \limits _{l=1}^{n}\left| z_{1}\right| ^{l-1}\left| z_{2}\right| ^{n-l}<n\ \ \ \left( n\in \mathbb {N} _{k}\right) . \end{aligned}$$

Hence, by (15) we have

$$\begin{aligned} \left| f\left( z_{1}\right) -f\left( z_{2}\right) \right|&\ge \left| h\left( z_{1}\right) -h\left( z_{2}\right) \right| -\left| g\left( z_{1}\right) -g\left( z_{2}\right) \right| \\&\ge \left| z_{1}-z_{2}-\sum \limits _{n=k}^{\infty }a_{n}\left( z_{1}^{n}-z_{2}^{n}\right) \right| -\left| \sum \limits _{n=k}^{\infty }\overline{b_{n}\left( z_{1}^{n}-z_{2}^{n}\right) }\right| \\&\ge \left| z_{1}-z_{2}\right| -\sum \limits _{n=k}^{\infty }\left| a_{n}\right| \left| z_{1}^{n}-z_{2}^{n}\right| -\sum \limits _{n=k}^{\infty }\left| b_{n}\right| \left| z_{1}^{n}-z_{2}^{n}\right| \\&=\left| z_{1}-z_{2}\right| \left( 1-\sum \limits _{n=k}^{\infty }\left| a_{n}\right| \left| \frac{z_{1}^{n}-z_{2}^{n}}{ z_{1}-z_{2}}\right| -\sum \limits _{n=k}^{\infty }\left| b_{n}\right| \left| \frac{z_{1}^{n}-z_{2}^{n}}{z_{1}-z_{2}} \right| \right) \\&>\left| z_{1}-z_{2}\right| \left( 1-\sum \limits _{n=k}^{\infty }n\left| a_{n}\right| -\sum \limits _{n=k}^{\infty }n\left| b_{n}\right| \right) \ge 0. \end{aligned}$$

This leads to the univalence of fi.e.$f\in \mathcal {S}_{ \mathcal {H}}.$ Therefore, $f\in \mathcal {S}^{m}(k;A,B)$ if and only if there exists a complex-valued function $\omega {,\ }\omega (0)=0,$$\left| \omega (z)\right| <1\,\left( z\in \mathbb {U}\right)$ such that

$$\begin{aligned} \frac{D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) }{D_{ \mathcal {H}}^{m}f\left( z\right) }=\frac{1+A\omega (z)}{1+B\omega (z)}\ \ \ \left( z\in \mathbb {U}\right) , \end{aligned}$$

or equivalently

$$\begin{aligned} \left| \frac{D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -D_{\mathcal {H}}^{m}f\left( z\right) }{BD_{\mathcal {H}}\left( D_{ \mathcal {H}}^{m}f\right) \left( z\right) -AD_{\mathcal {H}}^{m}f\left( z\right) }\right| <1\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

(16)

Thus, it is suffice to prove that

$$\begin{aligned} \left| D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -D_{\mathcal {H}}^{m}f\left( z\right) \right| -\left| BD_{\mathcal {H} }\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -AD_{\mathcal {H} }^{m}f\left( z\right) \right| <0\,\,\,(z\in \mathbb {U}_{0}). \end{aligned}$$

Indeed, letting $\left| z\right| =r\,~(0<r<1)$ we have

$$\begin{aligned}&\left| D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -D_{\mathcal {H}}^{m}f\left( z\right) \right| -\left| BD_{ \mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -AD_{\mathcal {H}}^{m}f\left( z\right) \right| \\&\quad =\left| \sum \limits _{n=k}^{\infty }\left( n-1\right) \lambda _{n}a_{n}z^{n}-\left( -1\right) ^{n}\sum \limits _{n=k}^{\infty }\left( n+1\right) \lambda _{n}\overline{b_{n}}\overline{z}^{n}\right| \\&\qquad -\left| \left( B-A\right) z+\sum \limits _{n=k}^{\infty }\left( Bn-A\right) \lambda _{n}a_{n}z^{n}+\left( -1\right) ^{n}\sum \limits _{n=k}^{\infty }\left( Bn+A\right) \lambda _{n}\overline{b_{n} }\overline{z}^{n}\right| \\&\quad \le \sum \limits _{n=k}^{\infty }\left( n-1\right) \lambda _{n}\left| a_{n}\right| r^{n}+\sum \limits _{n=k}^{\infty }\left( n+1\right) \lambda _{n}\left| b_{n}\right| r^{n}-\left( B-A\right) r \\&\qquad +\sum \limits _{n=k}^{\infty }\left( Bn-A\right) \lambda _{n}\left| a_{n}\right| r^{n}+\sum \limits _{n=k}^{\infty }\left( Bn+A\right) \lambda _{n}\left| b_{n}\right| r^{n} \\&\quad \le r\left\{ \sum \limits _{n=k}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) r^{n-1}-\left( B-A\right) \right\} <0. \end{aligned}$$

whence $f\in \mathcal {S}_{\mathcal {H}}^{m}(k;A,B)$. $\square$

The next theorem, shows that the condition (12) is also the sufficient condition for a function $f\in \mathcal {H}$ of correlated coefficients to be in the class $\mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B).$

Theorem 8

Let $f\in \mathcal {T}^{m}\left( k,\eta \right)$ be a function of the form (4). Then $f\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ if and only if the condition (12) holds true.

Proof

In view of Theorem 7 we need only show that each function $f\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ satisfies the coefficient inequality (12). If $f\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B),$ then it is of the form (4) with (9) and it satisfies (16 ) or equivalently

$$\begin{aligned} \left| \frac{\sum \nolimits _{n=k}^{\infty }\left( n-1\right) \lambda _{n}a_{n}z^{n}-\left( -1\right) ^{n}\sum \nolimits _{n=k}^{\infty }\left( n+1\right) \lambda _{n}\overline{b_{n}}\overline{z}^{n}}{\left( B-A\right) z+\sum \nolimits _{n=k}^{\infty }\left( Bn-A\right) \lambda _{n}a_{n}z^{n}+\left( -1\right) ^{n}\sum \nolimits _{n=k}^{\infty }\left( Bn+A\right) \lambda _{n}\overline{b_{n}}\overline{z}^{n}}\right| <1\ \ \,\,(z\in \mathbb {U}). \end{aligned}$$

Therefore, putting $z=re^{i\eta }~(0\le r<1)$ by (10) and (9) we obtain

$$\begin{aligned} \frac{\sum \nolimits _{n=k}^{\infty }\left( n-1\right) \lambda _{n}\left| a_{n}\right| +\left( n+1\right) \lambda _{n}\left| b_{n}\right| r^{n-1}}{\left( B-A\right) -\sum \nolimits _{n=k}^{\infty }\left\{ \left( Bn-A\right) \lambda _{n}\left| a_{n}\right| +\left( Bn+A\right) \lambda _{n}\left| b_{n}\right| \right\} r^{n-1}}<1. \end{aligned}$$

(17)

It is clear that the denominator of the left hand side cannot vanish for $r\in \left\langle 0,1\right) .$ Moreover, it is positive for $r=0,$ and in consequence for $r\in \left\langle 0,1\right) .$ Thus, by (17) we have

$$\begin{aligned} \sum \limits _{n=k}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) r^{n-1}<B-A\ \ \ (0\le r<1). \end{aligned}$$

(18)

The sequence of partial sums $\left\{ S_{n}\right\}$ associated with the series $\sum \nolimits _{n=k}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right)$ is nondecreasing sequence. Moreover, by (18) it is bounded by $B-A.$ Hence, the sequence$\ \left\{ S_{n}\right\}$ is convergent and

$$\begin{aligned} \sum \limits _{n=k}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) =\lim _{n\rightarrow \infty }S_{n}\le B-A, \end{aligned}$$

which yields the assertion (12). $\square$

The following result may be proved in much the same way as Theorem 8.

Theorem 9

Let $f\in \mathcal {H}$ be a function of the form (4). Then $f\in \mathcal {R}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ if and only if

$$\begin{aligned} \sum \limits _{n=k}^{\infty }\lambda _{n}\left( \left| a_{n}\right| +\left| b_{n}\right| \right) \le \frac{B-A}{1+B}. \end{aligned}$$

By Theorems 8 and 9 we have the following corollary.

Corollary 1

Let

$$\begin{aligned} \phi \left( z\right)= & {} z+\sum \limits _{n=k}^{\infty }\left( \frac{1}{n-a} z^{n}+\frac{1}{n+a}\overline{z}^{n}\right) \ \ \ \left( z\in \mathbb {U},\ a= \frac{1+A}{1+B}\right) , \\ \omega \left( z\right)= & {} z+\sum \limits _{n=k}^{\infty }\left( \left( n-a\right) z^{n}+\left( n+a\right) \overline{z}^{n}\right) \ \ \ \left( z\in \mathbb {U},\ a=\frac{1+A}{1+B}\right) . \nonumber \end{aligned}$$

(19)

Then

$$\begin{aligned} f\in & {} \mathcal {R}_{\mathcal {T}}^{m}(k,\eta ;A,B)\Leftrightarrow f*\phi \in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B), \\ f\in & {} \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)\Leftrightarrow f*\omega \in \mathcal {R}_{\mathcal {T}}^{m}(k,\eta ;A,B). \end{aligned}$$

In particular,

$$\begin{aligned} \mathcal {R}_{\mathcal {T}}^{1}(k,\eta ;-1,B)=\mathcal {S}_{\mathcal {T} }^{0}(k,\eta ;-1,B). \end{aligned}$$

4 Topological properties

We consider the usual topology on $\mathcal {H}$ defined by a metric in which a sequence $\left\{ f_{n}\right\}$ in $\mathcal {H}$ converges to f if and only if it converges to f uniformly on each compact subset of $\mathbb {U}$. It follows from the theorems of Weierstrass and Montel that this topological space is complete.

Let $\mathcal {F}$ be a subclass of the class ${\mathcal {H}}$. A functions $f\in \mathcal {F}$ is called an extreme point of$\mathcal {F}$ if the condition

$$\begin{aligned} f=\gamma f_{1}+\left( 1-\gamma \right) f_{2}\quad \left( f_{1},f_{2}\in \mathcal {F},\;0<\gamma <1\right) \end{aligned}$$

implies $f_{1}=f_{2}=f.$ We shall use the notation $E\mathcal {F}$ to denote the set of all extreme points of $\mathcal {F}.$ It is clear that $E\mathcal {F }\subset \mathcal {F}.$

We say that $\mathcal {F}$ is locally uniformly bounded if for each $r,\,0<r<1,$ there is a real constant $M=M\left( r\right)$ so that

$$\begin{aligned} \left| f(z)\right| \le M\quad \left( f\in \mathcal {F},\;\left| z\right| \le r\right) . \end{aligned}$$

We say that a class $\mathcal {F}$ is convex if

$$\begin{aligned} \gamma f+(1-\gamma )g\in \mathcal {F}\quad \quad (f,g\in \mathcal {F},\,0\le \gamma \le 1). \end{aligned}$$

Moreover, we define the closed convex hull of $\mathcal {F}$ as the intersection of all closed convex subsets of ${\mathcal {H}}$ that contain $\mathcal {F}$. We denote the closed convex hull of $\mathcal {F}$ by $\overline{co}\mathcal {F}.$

A real-valued functional $\mathcal {J}:{\mathcal {H}}\rightarrow \mathbb {R}$ is called convex on a convex class $\mathcal {F}\subset {\mathcal {H}}$ if

$$\begin{aligned} \mathcal {J}\left( \gamma f+\left( 1-\gamma \right) g\right) \le \gamma \mathcal {J}\left( f\right) +\left( 1-\gamma \right) \mathcal {J}\left( g\right) \quad \left( f,g\in \mathcal {F},\;0\le \gamma \le 1\right) . \end{aligned}$$

The Krein–Milman theorem (see [14]) is fundamental in the theory of extreme points. In particular, it implies the following lemma.

Lemma 1

[6, pp.45] Let $\mathcal {F}$ be a non-empty compact convex subclass of the class $\mathcal {H}$ and $\mathcal {J}:\mathcal {H}\rightarrow \mathbb {R}$ be a real-valued, continuous and convex functional on $\mathcal {F }.$ Then

$$\begin{aligned} \max \left\{ \mathcal {J}(f):f\in \mathcal {F}\right\} =\max \left\{ \mathcal {J }(f):f\in E\mathcal {F}\right\} . \end{aligned}$$

Since $\mathcal {H}$ is a complete metric space, Montel’s theorem (see [16]) implies the following lemma.

Lemma 2

A class $\mathcal {F}\subset \mathcal {H}$ is compact if and only if $\mathcal {F}$ is closed and locally uniformly bounded.

Theorem 10

The class $\mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ is convex and compact subset of $\mathcal {H}$.

Proof

Let $f_{1},f_{2}\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ be functions of the form (7),${\;0\le \gamma \le 1.}$ Since

$$\begin{aligned} \gamma f_{1}(z)+(1-\gamma )f_{2}\left( z\right) =z+\sum \limits _{n=k}^{\infty }\left\{ \left( \gamma a_{1,n}+\left( 1-\gamma \right) a_{2,n}\right) z^{n}+ \overline{\left( \gamma b_{1,n}+\left( 1-\gamma \right) b_{2,n}\right) z^{n}} \right\} , \end{aligned}$$

and by Theorem 8 we have

$$\begin{aligned}&\sum \limits _{n=k}^{\infty }\left\{ \alpha _{n}\left| \gamma a_{1,n}+\left( 1-\gamma \right) a_{2,n}\right| +\beta _{n}\left| \gamma b_{1,n}+\left( 1-\gamma \right) b_{2,n}z^{n}\right| \right\} \\&\quad \le \gamma \sum \limits _{n=k}^{\infty }\left\{ \alpha _{n}\left| a_{1,n}\right| +\beta _{n}\left| b_{1,n}\right| \right\} +\left( 1-\gamma \right) \sum \limits _{n=k}^{\infty }\left\{ \alpha _{n}\left| a_{2,n}\right| +\beta _{n}\left| b_{2,n}\right| \right\} \\&\quad \le \gamma \left( B-A\right) +\left( 1-\gamma \right) \left( B-A\right) =B-A, \end{aligned}$$

the function ${\phi }=\gamma f_{1}+(1-\gamma )f_{2}$ belongs to the class $\mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$. Hence, the class is convex. Furthermore, for $f\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B),\ \left| z\right| \le r,\;0<r<1,$ we have

$$\begin{aligned} \left| f(z)\right| \le r+\sum \limits _{n=k}^{\infty }\left( \left| a_{n}\right| +\left| b_{n}\right| \right) r^{n}\le r+\sum \limits _{n=k}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) \le r+\left( B-A\right) . \end{aligned}$$

(20)

Thus, we conclude that the class $\mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ is locally uniformly bounded. By Lemma 2, we only need to show that it is closed i.e. if $f_{l}\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B){\;\;}\left( l\in \mathbf {\mathbb {N}}\right)$ and $f_{l}\rightarrow f,$ then $f\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B).$ Let $f_{l}$ and f are given by (7) and (4), respectively. Using Theorem 8 we have

$$\begin{aligned} \sum \limits _{n=k}^{\infty }\left( \left| \alpha _{n}a_{l,n}\right| +\left| \beta _{n}b_{l,n}\right| \right) \le B-A\;\;\left( l\in \mathbf {\mathbb {N}}\right) .\, \end{aligned}$$

(21)

Since $f_{l}\rightarrow f,$ we conclude that $\left| a_{l,n}\right| \rightarrow \left| a_{n}\right|$ and $\left| b_{l,n}\right| \rightarrow \left| b_{n}\right|$ as $l\rightarrow \infty \;\left( n\in \mathbf {\mathbb {N}}\right)$. The sequence of partial sums $\left\{ S_{n}\right\}$ associated with the series $\sum \nolimits _{n=k}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta \left| _{n}b_{n}\right| \right)$ is nondecreasing sequence. Moreover, by (21) it is bounded by $B-A.$ Therefore, the sequence$\ \left\{ S_{n}\right\}$ is convergent and

$$\begin{aligned} \sum \limits _{n=k}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) =\lim _{n\rightarrow \infty }S_{n}\le B-A. \end{aligned}$$

This gives the condition (12), and, in consequence, $f\in \mathcal {S} _{\mathcal {T}}^{m}(k,\eta ;A,B),$ which completes the proof. $\square$

Theorem 11

$$\begin{aligned} E\mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)=\left\{ h_{n}:\;n\in \mathbb {N} _{k-1}\right\} \cup \left\{ g_{n}:\;n\in \mathbb {N}_{k}\right\} , \end{aligned}$$

where

$$\begin{aligned} h_{k-1}(z)=z,\ h_{n}(z)=z-\frac{B-A}{\alpha _{n}{e^{i\left( n-1\right) \eta } }}z^{n},\;g_{n}(z)=z+\frac{B-A}{\beta _{n}{e^{i\left( 1-n\right) \eta }}} \overline{z}^{n}\ \ (z\in \mathbb {U}). \end{aligned}$$

(22)

Proof

Suppose that $0<\gamma <1$ and

$$\begin{aligned} g_{n}=\gamma f_{1}+\left( 1-\gamma \right) f_{2}, \end{aligned}$$

where $f_{1},f_{2}\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ are functions of the form (7). Then, by (12) we have $\left| b_{1,n}\right| =\left| b_{2,n}\right| =$$\frac{B-A}{\beta _{n}} ,$ and, in consequence, $a_{1,l}=a_{2,l}=0$ for $l\in \mathbb {N}_{k}$ and $b_{1,l}=b_{2,l}=0$ for $l\in \mathbb {N}_{k}\diagdown \left\{ n\right\} .$ It follows that $g_{n}=f_{1}=f_{2},$ and consequently $g_{n}\in E\mathcal {S}_{ \mathcal {T}}^{*}(k,\eta ;A,B).$ Similarly, we verify that the functions $h_{n}$ of the form (22) are the extreme points of the class $\mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B).$ Now, suppose that a function f belongs to the set $E\mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ and f is not of the form (22). Then there exists $m\in \mathbb {N}_{k}$ such that

$$\begin{aligned} 0<\left| a_{m}\right|<\frac{B-A}{{\alpha }_{m}}\text { or } 0<\left| b_{m}\right| <\frac{B-A}{{\beta }_{m}}. \end{aligned}$$

If $0<\left| a_{m}\right| <\frac{B-A}{{\alpha }_{m}}$, then putting

$$\begin{aligned} \gamma =\frac{{\alpha }_{m}\left| a_{m}\right| }{B-A},~\varphi = \frac{1}{1-\gamma }\left( f-\gamma h_{m}\right) , \end{aligned}$$

we have that $0<\gamma <1,\ h_{m}\ne \varphi$ and

$$\begin{aligned} f=\gamma h_{m}+\left( 1-\gamma \right) \varphi . \end{aligned}$$

Thus, $f\notin E\mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B).$ Similarly, if $0<\left| b_{m}\right| <\frac{B-A}{\beta _{n}}$, then putting

$$\begin{aligned} \gamma =\frac{{\beta }_{m}\left| b_{m}\right| }{B-A},~\phi =\frac{1}{ 1-\gamma }\left( f-\gamma g_{m}\right) , \end{aligned}$$

we have that $0<\gamma <1,\ g_{m}\ne \phi$ and

$$\begin{aligned} f=\gamma g_{m}+\left( 1-\gamma \right) \phi . \end{aligned}$$

It follows that $f\notin E\mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B),$ and the proof is completed. $\square$

5 Applications

It is clear that if the class

$$\begin{aligned} \mathcal {F}=\left\{ f_{n}\in {\mathcal {H}}:\;n\in \mathbf { \mathbb {N} }\right\} , \end{aligned}$$

is locally uniformly bounded, then

$$\begin{aligned} \overline{co}\mathcal {F}=\left\{ \sum _{n=1}^{\infty }\gamma _{n}f_{n}:\;\;\;\sum _{n=1}^{\infty }\gamma _{n}=1,\;\gamma _{n}\ge 0\;\left( n\in \mathbf { \mathbb {N} }\right) \right\} . \end{aligned}$$

(23)

Thus, by Theorem 6 we have the following corollary.

Corollary 2

$$\begin{aligned} \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)=\left\{ \sum _{n=k-1}^{\infty }\left( \gamma _{n}h_{n}+\delta _{n}g_{n}\right) :\;\sum _{n=k-1}^{\infty }\left( \gamma _{n}+\delta _{n}\right) =1\;\left( \delta _{k-1}=0,\gamma _{n},\delta _{n}\ge 0\right) \right\} , \end{aligned}$$

where $h_{n},g_{n}$ are defined by (22).

For each fixed value of $m,n\in \mathbf { \mathbb {N} }_{k},\;z\in \mathbb {U},$ the following real-valued functionals are continuous and convex on ${\mathcal {H}}$:

$$\begin{aligned} \mathcal {J}\left( f\right) =\left| a_{n}\right| ,\_\mathcal {J}\left( f\right) =\left| b_{n}\right| ,\;\mathcal {J}\left( f\right) =\left| f\left( z\right) \right| \;\mathcal {J}\left( f\right) =\left| D_{\mathcal {H}}f\left( z\right) \right| {\quad \left( f\in { \mathcal {H}}\right) .\;} \end{aligned}$$

(24)

Moreover, for $\gamma \ge 1,\ 0<r<1,$ the real-valued functional

$$\begin{aligned} \mathcal {J}\left( f\right) =\left( \frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| f\left( re^{i\theta }\right) \right| ^{\gamma }d\theta \right) ^{1/\gamma }\ \ \ {\left( f\in {\mathcal {H}}\right) } \end{aligned}$$

(25)

is also continuous and convex on ${\mathcal {H}}.$

Therefore, by Lemma 1 and Theorem 6 we have the following corollaries.

Corollary 3

Let $f\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ be a function of the form (11). Then

$$\begin{aligned} \left| a_{n}\right| \le {\frac{B-A}{\alpha _{n}},\ }\left| b_{n}\right| \le {\frac{B-A}{\beta _{n}}}~~\,\,(n\in \mathbb {N} _{k}), \end{aligned}$$

(26)

where $\alpha _{n},\beta _{n}$ are defined by (13). The result is sharp. The functions $h_{n},g_{n}$ of the form (22) are the extremal functions.

Corollary 4

Let $f\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B)$, $\left| z\right| =r<1$. Then

$$\begin{aligned} \,r-\frac{B-A}{\lambda _{k}\left( k-1+kB-A\right) }r^{k}\le & {} \left| f(z)\right| \le r+\frac{B-A}{\lambda _{k}\left( k-1+kB-A\right) }r^{k},\\ \,r-\frac{k\left( B-A\right) }{\lambda _{k}\left( k-1+kB-A\right) }r^{k}\le & {} \left| D_{\mathcal {H}}f(z)\right| \le r+\frac{k\left( B-A\right) }{\lambda _{k}\left( k-1+kB-A\right) }r^{k}, \end{aligned}$$

where $\lambda _{k}$ is defined by (11). The result is sharp. The function $h_{k}$ of the form (22) is the extremal function.

Corollary 5

Let $0<r<1,\ \gamma \ge 1.$ If $f\in \mathcal {S}_{\mathcal {T} }^{m}(k,\eta ;A,B),$ then

$$\begin{aligned} \frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| f(re^{i\theta })\right| ^{\gamma }d\theta\le & {} \frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| h_{k}(re^{i\theta })\right| ^{\lambda }d\theta , \\ \frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| D_{\mathcal {H} }f(z)\right| ^{\gamma }d\theta\le & {} \frac{1}{2\pi }\int \limits _{0}^{2 \pi }\left| D_{\mathcal {H}}h_{k}(re^{i\theta })\right| ^{\gamma }d\theta , \end{aligned}$$

where $h_{k}$ is the function defined by (22).

The following covering result follows from Corollary 4.

Corollary 6

If $f\in \mathcal {S}_{\mathcal {T}}^{m}(k,\eta ;A,B),$ then $\mathbb {U}\left( r\right) \subset f\left( \mathbb {U}\right) ,$ where

$$\begin{aligned} \,r=1-\frac{B-A}{\lambda _{k}\left( k-1+kB-A\right) }. \end{aligned}$$

By using Corollary 1 and the results above we obtain corollaries listed below.

Corollary 7

The class $\mathcal {R}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ is convex and compact subset of $\mathcal {H}$. Moreover,

$$\begin{aligned} E\mathcal {R}_{\mathcal {T}}^{m}(k,\eta ;A,B)=\left\{ h_{n}:\;n\in \mathbb {N} _{k-1}\right\} \cup \left\{ g_{n}:\;n\in \mathbb {N}_{k}\right\} , \end{aligned}$$

and

$$\begin{aligned} \mathcal {R}_{\mathcal {T}}^{m}(k,\eta ;A,B)=\left\{ \sum _{n=k-1}^{\infty }\left( \gamma _{n}h_{n}+\delta _{n}g_{n}\right) :\;\sum _{n=k-1}^{\infty }\left( \gamma _{n}+\delta _{n}\right) =1\;\left( \delta _{k-1}=0,\gamma _{n},\delta _{n}\ge 0\right) \right\} , \end{aligned}$$

where

$$\begin{aligned} h_{k-1}(z)=z,\ h_{n}(z)=z-\frac{\left( B-A\right) {e^{i\left( 1-n\right) \eta }}}{\left( 1+B\right) \lambda _{n}}z^{n},\;g_{n}(z)=z+\frac{\left( B-A\right) {e^{i\left( n-1\right) \eta }}}{\left( 1+B\right) \lambda _{n}} \overline{z}^{n}\ \ (z\in \mathbb {U}). \end{aligned}$$

(27)

Corollary 8

Let $f\in \mathcal {R}_{\mathcal {T}}^{m}(k,\eta ;A,B)$ be a function of the form (4). Then

$$\begin{aligned} \left| a_{n}\right|&\le \frac{B-A}{\left( 1+B\right) \lambda _{n}} ,\left| b_{n}\right| \le \frac{B-A}{\left( 1+B\right) \lambda _{n}}~~\,\,(n\in \mathbb {N} _{k}), \\ r-&\frac{B-A}{\left( 1+B\right) \lambda _{k}}r^{k}\le \left| f(z)\right| \le r+\frac{B-A}{\left( 1+B\right) \lambda _{k}}r^{k}\ \ \ \left( \left| z\right| =r<1\right) , \\ r-&\frac{(B-A)k}{\left( 1+B\right) \lambda _{k}}r^{k}\le \left| D_{ \mathcal {H}}f(z)\right| \le r+\frac{\left( B-A\right) k}{\left( 1+B\right) \lambda _{k}}r^{k}\ \ \ \left( \left| z\right| =r<1\right) , \\&\frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| f(re^{i\theta })\right| ^{\gamma }d\theta \le \frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| h_{k}(re^{i\theta })\right| ^{\lambda }d\theta , \\&\frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| D_{\mathcal {H} }f(re^{i\theta })\right| ^{\gamma }d\theta \le \frac{1}{2\pi } \int \limits _{0}^{2\pi }\left| D_{\mathcal {H}}h_{k}(re^{i\theta })\right| ^{\gamma }d\theta , \end{aligned}$$

where $\lambda _{n}$ is defined by (11). The results are sharp. The functions $h_{n},g_{n}$ of the form (27) are the extremal functions.

Corollary 9

If $f\in \mathcal {R}_{\mathcal {T}}^{m}(k,\eta ;A,B),$ then $\mathbb {U}\left( r\right) \subset f\left( \mathbb {U}\right) ,$ where

$$\begin{aligned} \,r=1-\frac{B-A}{\left( 1+B\right) \lambda _{k}}. \end{aligned}$$

The classes $\mathcal {S}_{\mathcal {H}}^{n}(k;A,B)$ and $\mathcal {R}_{ \mathcal {H}}^{n}(k;A,B)$ are related to harmonic starlike functions, harmonic convex functions and harmonic Janowski functions.

The classes $\mathcal {S}_{\mathcal {H}}(\alpha ):=\mathcal {S}_{\mathcal {H} }^{0}(2;2\alpha -1,1)$ and $\mathcal {K}_{\mathcal {H}}(\alpha ):=\mathcal {S}_{ \mathcal {H}}^{1}(2;2\alpha -1,1)$ were investigated by Jahangiri [9] (see also [2, 18]). They are the classes of starlike and convex functions of order $\alpha ,$ respectively. The classes $N_{\mathcal {H} }(\alpha ):=\mathcal {R}_{\mathcal {H}}^{1}(2;2\alpha -1,1)$ and $R_{\mathcal {H }}(\alpha ):=\mathcal {R}_{\mathcal {H}}^{2}(2;2\alpha -1,1)$ were studied in [1] (see also [13]). Finally, the classes $\mathcal {S}_{\mathcal {H }}:=\mathcal {S}_{\mathcal {H}}(0)$ and $\mathcal {K}_{\mathcal {H}}:=\mathcal {K} _{\mathcal {H}}(0)$ are the classes of functions which are starlike and convex in ${\mathbb {U}}\left( r\right) ,$ respectively, for all $r\in \left( 0,1\right\rangle .$ We should notice, that the classes $\mathcal {S}(A,B):= \mathcal {S}_{\mathcal {H}}(2;A,B)\cap \mathcal {A}$ and $\mathcal {R}(A,B):= \mathcal {R}_{\mathcal {H}}(2;A,B)\cap \mathcal {A}$ were introduced by Janowski [12].

Using Theorems 1 or 2 to the classes defined above we obtain corollaries listed below (see also [6]).

Corollary 10

$$\begin{aligned} \mathcal {S}_{\mathcal {H}}(\alpha )=\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right) :=z\frac{2\left( 1-\alpha \right) \xi +\left( 1-\xi +2\alpha \xi \right) z}{\left( 1-z\right) ^{2}}-\overline{z}\frac{ 2+2\alpha \xi -\left( 1-\xi +2\alpha \xi \right) \overline{z}}{\left( 1- \overline{z}\right) ^{2}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Corollary 11

$$\begin{aligned} \mathcal {K}_{\mathcal {H}}(\alpha )=\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right) :=z\frac{\left( 1-\alpha \right) \xi +\left( 1+\alpha \xi \right) z}{\left( 1-z\right) ^{3}}+\overline{z}\frac{1+\alpha \xi +\left( 1-\alpha \right) \xi \overline{z}}{\left( 1-\overline{z}\right) ^{3}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Corollary 12

$$\begin{aligned} N_{\mathcal {H}}(\alpha )=\left\{ \delta _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \delta _{\xi }\left( z\right) :=z\frac{2\left( 1-\alpha \right) \xi -\left( 2\alpha \xi -\xi +1\right) \left( z^{2}-2z\right) }{\left( 1-z\right) ^{2}}- \overline{z}\frac{1+\xi }{\left( 1-\overline{z}\right) ^{2}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Corollary 13

$$\begin{aligned} \mathcal {S}_{\mathcal {H}}=\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right) :=z\frac{2\xi +\left( 1-\xi \right) z}{\left( 1-z\right) ^{2}}-\overline{z}\frac{2-\left( 1-\xi \right) \overline{z}}{ \left( 1-\overline{z}\right) ^{2}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

Corollary 14

$$\begin{aligned} \mathcal {K}_{\mathcal {H}}=\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right) :=z\frac{\xi +z}{\left( 1-z\right) ^{3}}+ \overline{z}\frac{1+\xi \overline{z}}{\left( 1-\overline{z}\right) ^{3}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

The class $\mathcal {S}_{\mathcal {H}}^{n}(k;A,B)$ generalize also classes of starlike functions of complex order. The class ${\mathcal {CS}}_{\mathcal {H} }(\gamma ):=\mathcal {S}_{\mathcal {H}}\left( 2;1-2\gamma ,1\right) \ \ \ \left( \gamma \in \mathbb {C}\smallsetminus \left\{ 0\right\} \right)$ was defined by Yalçin and Öztürk [24]. In particular, if we put $\gamma :=\frac{1-\alpha }{1+e^{i\eta }},$ then we obtain the class ${\mathcal {RS}}_{\mathcal {H}}(\alpha ,\eta ):=\mathcal {S}_{\mathcal {H}}\left( 2; \frac{2\alpha -1+e^{i\eta }}{1+e^{i\eta }},1\right)$ studied by Yalçin et al. [25]. It is the class of functions $f\in \mathcal {H} _{0}$ such that

$$\begin{aligned} \mathrm {Re}\left\{ \left( 1+e^{i\eta }\right) \frac{D_{\mathcal {H}}f\left( z\right) }{f\left( z\right) }-e^{i\eta }\right\} >\alpha \ \ \ \left( z\in \mathbb {U},\ \eta \in \mathbb {R}\right) . \end{aligned}$$

Thus, by Theorem 4 we have the following two corollaries.

Corollary 15

$$\begin{aligned} {\mathcal {CS}}_{\mathcal {H}}(\gamma )=\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right) :=z\frac{2\gamma \xi +\left( 1+\xi -2\gamma \xi \right) z}{\left( 1-z\right) ^{2}}-\overline{z}\frac{2+2\left( 1-\gamma \right) \xi -\left( 1+\xi -2\gamma \xi \right) \overline{z}}{\left( 1- \overline{z}\right) ^{2}}\ \ \ \left( z\in \mathbb {U}\right) . \end{aligned}$$

(28)

Corollary 16

$$\begin{aligned} {{{\mathcal {CS}}}}_{\mathcal {H}}^{n}(\alpha ,\eta )=\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where $\psi _{\xi }$ is defined by (28) with $\gamma :=\frac{ 1-\alpha }{1+e^{i\eta }}.$

Remark 1

By choosing the parameters in the defined classes of functions we can obtain new and also well-known results (see for example [1,2,3, 5,6,7,8,9,10,11,12,13, 18, 21,22,25]) .