Abstract
Let f be analytic in the unit disk \({\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}\), and \({{\mathcal {S}}}\) be the subclass of normalized univalent functions given by \(f(z)=z+\sum _{n=2}^{\infty }a_n z^n\) for \(z\in {\mathbb {D}}\). We give sharp bounds for the modulus of the second Hankel determinant \( H_2(2)(f)=a_2a_4-a_3^2\) for the subclass \( {\mathcal F_{O}}(\lambda ,\beta )\) of strongly Ozaki close-to-convex functions, where \(1/2\le \lambda \le 1\), and \(0<\beta \le 1\). Sharp bounds are also given for \(|H_2(2)(f^{-1})|\), where \(f^{-1}\) is the inverse function of f. The results settle an invariance property of \(|H_2(2)(f)|\) and \(|H_2(2)(f^{-1})|\) for strongly convex functions.
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1 Introduction
Let \({{\mathcal {A}}}\) denote the class of analytic functions f in the unit disk \({\mathbb {D}}:=\{ z\in {\mathbb {C}}: |z|<1 \}\) normalized by \(f(0)=0=f'(0)-1\). Then, for \(z\in {\mathbb {D}}\), \(f\in {{\mathcal {A}}}\) has the following representation
Let \({{\mathcal {S}}}\) denote the subclass of all univalent (i.e., one-to-one) functions in \({{\mathcal {A}}}\).
Denote by \({{\mathcal {S}}}^*\) the subclass of \({{\mathcal {S}}}\) consisting of starlike functions, i.e., functions f which map \({\mathbb {D}}\) onto a set which is star-shaped with respect to the origin. Then, it is well known that a function \(f \in {{\mathcal {S}}}^*\) if, and only if, for \(z\in {\mathbb {D}},\)
and that \({\mathcal {S}}^*\subset {\mathcal {S}}.\)
Next denote by \({{\mathcal {K}}}\) the subclass of \({{\mathcal {S}}}\) consisting of functions which are close-to-convex, i.e., functions f which map \({\mathbb {D}}\) onto a close-to-convex domain, if, and only if, there exist \(0\le \delta \le 2\pi \) and \(g\in {\mathcal {S}}^*\) such that for \(z\in {\mathbb {D}},\)
Again it is well known that \({\mathcal {K}}\subset {\mathcal {S}}\).
The class \({\mathcal {C}}(\alpha )\) for \(0\le \alpha <1\) of convex functions of order \(\alpha \) consisting of functions \(f\in {\mathcal {A}}\) satisfying
for \(z\in {\mathbb {D}},\) is also well known and has been widely studied. When \(\alpha =0\), we obtain the class \({\mathcal {C}}\) of convex functions.
Little attention has been given to the case when \(\alpha <0\), but several authors have considered the class \({\mathcal {C}}(-1/2)\), consisting of functions \(f\in {\mathcal {A}}\) satisfying
for \(z\in {\mathbb {D}}\), whose members are known to be close-to-convex, and so univalent.
The class \({\mathcal {F}}_O(\lambda )\), defined for \(1/2\le \lambda \le 1\) by
for \(z\in {\mathbb {D}}\), was formally introduced in [1] and is also known to be a subclass of the close-convex functions. More information concerning the coefficients of functions in \({\mathcal {F}}_O(\lambda )\) (called Ozaki close-to-convex functions), has also be found by Allu, Lecko and Thomas (to appear). We note that clearly \({\mathcal {F}}_O(1/2)={\mathcal {C}}\), and \({\mathcal {F}}_O(1)={\mathcal {C}}(-1/2)\).
Also in [1], the following notion of strongly Ozaki functions was introduced, and some basic properties were obtained.
Definition 1
Let \(0< \beta \le 1\) and \(1/2\le \lambda \le 1\). Then, \(f\in {\mathcal {A}}\) is called a strongly Ozaki close-to-convex if, and only if, for \(z\in {\mathbb {D}},\)
We denote this class of functions by \({\mathcal {F}}_{O}(\lambda ,\beta )\), noting that when \(\beta =1\) this reduces to (3), and when \(\lambda =1/2\) we obtain the class \({\mathcal {C}}^{\beta }\) of strongly convex functions considered in [9].
For \(q,n \in {\mathbb {N}},\) the Hankel determinant \(H_{q,n}(f)\) of function \(f \in {{\mathcal {A}}}\) of the form (1) is defined as
Hankel matrices are used in the theory of Markov processes, of non-stationary signals, in Hamburger moment problem, and many other issues in both pure mathematics and technical applications.
To find the upper bound of \(H_{q,n}(f)\) for the whole class \({\mathcal {S}}\) of univalent functions as well as for its subclasses is an interesting problem to study. For the class \({\mathcal {S}}\), a basic result was shown by Pommerenke [7], who found the growth of \(H_{q,n}(f)\) dependent on q and n. In recent years there has been a great deal of attention devoted to finding bounds for the modulus of the second Hankel determinant \(H_2(2)(f)=a_2a_4-a_3^3,\) when f belongs to various subclasses of \({\mathcal {A}}\). Most authors have used the method introduced in [4], where the sharp bound \(|H_2(2)(f)|\le 1\) was found for \(f\in {\mathcal {S}}^*\). However, finding the sharp bound for \(|H_2(2)(f)|\) when \(f\in {\mathcal {K}}\) and even when \(\delta =0\) remains an open problem.
In [2], a sharp bound was obtained for \(|H_2(2)(f)|\) when \(f\in {\mathcal {F}}_O(\lambda )\). The main purpose of this paper is to give the sharp bounds for \( |H_2(2)(f)|\), when \(f\in {\mathcal {F}}_O(\lambda ,\beta )\), together with sharp bounds for \( |H_2(2)(f^{-1})|\), where \(f^{-1}\) is the inverse function of f.
2 Preliminary lemmas
Denote by \({{\mathcal {P}}}\), the class of analytic functions p in \({\mathbb {D}}\) with positive real part on \({\mathbb {D}}\) given by
We will use the following properties for the coefficients of functions \({{\mathcal {P}}}\), given by (5).
Lemma 1
[5, 6] If \(p\in {\mathcal {P}}\) and is given by (5) with \(c_1\ge 0\), then for some complex valued \(\zeta \) with \(|\zeta |\le 1\), and some complex valued \(\eta \) with \(|\eta | \le 1\),
Lemma 2
[3] Let \(\overline{{\mathbb {D}}} := \{z\in {\mathbb {C}}:|z|\le 1\}\), and for real numbers A, B, C, let
If \(AC\ge 0,\) then
If \(AC<0,\) then
where
We also note that from (4), we can write, for \(z\in {\mathbb {D}},\)
for some \(p\in {\mathcal {P}}\) and so equating coefficients we have
3 \(H_2(2)(f)\) for strongly Ozaki functions
We prove the following.
Theorem 1
Let \(\beta \in (0,1]\) and \(\lambda \in [1/2,1]\). If \(f\in {{\mathcal {F}}}_{O}(\lambda ,\beta )\), then
The inequalities are sharp.
Proof
From (11) we have
where
Since both the class \({{\mathcal {F}}}_{O}(\lambda ,\beta )\) and the functional are rotationally invariant, without loss of generality we may assume that \(c = c_1 \in [0,2]\). By Lemma 1, we obtain
for some \(\zeta \), \(\eta \in \overline{{\mathbb {D}}}\).
Since \(|\eta |\le 1\) and \(|\zeta |\le 1,\) by applying the triangle inequality to (13), we have
where for \(x\in [0,4],\)
with
I. Consider first the case \((5+2\lambda )\beta \le 2\).
Since \(b_1 \le 0\), and since \(\beta \le 2/(5+2\lambda )\) and \(1/2 \le \lambda \le 1\), the inequalities
hold. Thus, we get
Hence, \(G_1'(x) = 2b_1 +2b_2x \le 0\) holds for all \(x\in [0,4]\), and so \(G_1\) is decreasing on [0, 4]. Since \(c\in [0,2]\), it follows from (14) that
which establishes the first inequality in Theorem 1.
We next divide the case \((5+2\lambda )\beta >2\) into two cases.
II(a). First suppose that \(\beta ^2\lambda (1+2\lambda ) \ge 1\). Then, we see that \(b_1>0\) and \(b_2<0\). Indeed,
and
since \(\beta ^2 \lambda (1+2\lambda ) \le 3\) and \((5+2\lambda )\beta >2\). Moreover, by putting
we see that
holds, since
Therefore, \(G_1'(\tau )=0\) and \(G_1''(\tau )<0\) imply that for \(x\in [0,4],\)
and so from (12) and (14), we obtain the second inequality in Theorem 1 in this case.
II(b). Now assume that \(\beta ^2\lambda (1+2\lambda ) < 1\) and \((5+2\lambda )\beta >2\). By (13), when \(c=2\), we have
and, when \(c=0\), since \(\zeta \in \overline{{\mathbb {D}}}\), we also have
Now let \(c \in (0,2)\). Then, since \(\eta \in \overline{{\mathbb {D}}}\), we obtain
where
with
Noting that \(AC<0\) in this case, we now apply Lemma 2.
Simple calculations show that the first two alternatives when \(AC<0\) are not satisfied, and so we must consider R(A, B, C).
We first show that \(|C|(|B|+4|A|) > |AB|\) holds for all \(c\in (0,2)\). A calculation shows that
where for \(x\in [0,4],\)
with
and
Note that \(k_2 \ge 0\) implies \(k_1>0\). Indeed, since \(\beta ^2 \lambda (1+2\lambda ) <1\), we have \(3-2\beta ^2\lambda (1+2\lambda )>0\). Therefore, if \(k_2 \ge 0\), then
Thus, when \(k_2 \ge 0\), from \(k_0>0\) and \(k_1>0\), we have \(H_1(x)>0\), for \(x\in [0,4]\).
When \(k_2<0\), since \(H_1''\equiv 2k_2<0\), we get for \(x\in [0,4],\)
Next fix \(\lambda \in [1/2,1]\). Let \(I:=\left( 2/(5+2\lambda ),1/\sqrt{\lambda (1+2\lambda )}\right) \) and \(\varphi :I\rightarrow {\mathbb {R}}\) be defined by
Since
so \(\varphi \) is decreasing on I and for \(x\in I,\)
Thus, \(H_1(4) = k_0+8k_1+16k_2 = 16 \varphi (\beta ) >0\). Also, since \(H_1(0)=k_0>0\), by (17), \(H_1(x)>0\) for all \(x\in [0,4]\), which implies that \(|C|(|B|+4|A|) > |AB|\).
Next a calculation gives
where for \(x\in [0,4],\)
with
and
Note that \(l_2<0\), and let
be the zeroes of \(H_2\). Since \(l_2<0\), so \(\xi _1<\xi _2\). Moreover, since \(l_0>0\), we have \(-l_1+\sqrt{l_1^2-l_0l_2} >0\) and \(\xi _1<0\). Also \(\xi _2 < 4\), since this is equivalent to
which is true. In fact, a computation shows that
where \(t =\beta (5+2\lambda )\) and \(u=\beta ^2\lambda (1+2\lambda )\). From the inequalities \(t>0\) and \(0<u<1\), we obtain the inequality (18). Therefore, since \(l_2<0\), \(H_2(x) \ge 0\) holds for \(x\in (0,\xi _2]\), and \(H_2(x) \le 0\) holds for \(x\in [\xi _2,4)\). Thus, \(|AB| \le |C|(|B|-4|A|)\) holds when \(c\in (0,\sqrt{\xi _2}]\).
Thus, when \(c\in (0,\sqrt{\xi _2}]\), Lemma 2 gives
where for \(x\in [0,4],\)
with
Note that \(d_1>0\) and \(d_2<0\). We also note that \(G_2'(x)=0\) holds only for
Clearly, \(\tau >0\). Now we claim \(\tau < \xi _2\). Since
the inequality \(\tau < \xi _2\) is equivalent to
Moreover, by putting \(t =\beta (5+2\lambda )\) and \(u=\beta ^2\lambda (1+2\lambda )\) again, we get
Since \(t>0\) and \(0<u<1\), all terms in the right side in the above are positive, and the inequality (22) follows. So, \(\tau < \xi _2\) and, since \(d_2<0\), the function \(G_2\) has its maximum at \(\tau \) in \((0,\xi _2]\). Therefore, by (19), we have
Thus, using (12), the second inequality in Theorem 1 holds.
We are therefore left to consider the interval \(c\in [\sqrt{\xi _2},2)\), where Lemma 2 gives
and where for \(x\in [0,4],\)
and
It is easily checked that \(g_1\) and \(g_2\) are both decreasing on \([\xi _2,2)\). Hence, from (23), we have
where \(G_2\) is defined by (20). Since \(G_2(\xi _2) \le G_2(\tau )\), it follows from (12) and (24) that the second inequality in Theorem 1 holds once more.
In order to show that the inequalities are sharp, first let \(f_1\) be defined for \(z\in {\mathbb {D}}\) by
where \(p_1(z) := (1+z^2)/(1-z^2)\) for \(z\in {\mathbb {D}}.\) Then, \(f_1 \in {{\mathcal {F}}}_{O}(\lambda ,\beta )\) with
for \(z\in {\mathbb {D}}.\) Thus, the first bound in Theorem 1 is sharp when \((5+2\lambda )\beta \le 2\).
Next let \(f_2\) be defined for \(z\in {\mathbb {D}}\) by
where
with
Here \(\tau \) is defined by (15) or by (21), and in the first case \(0<\tau <4\) in view of (16), or in the second case, \(0<\tau<\xi _2<4\) as was shown in II(b). Therefore, \(b\in (0,2)\) and \(p_2\in {\mathcal {P}}.\) Thus, \(f_2 \in {{\mathcal {F}}}_{O}(\lambda ,\beta )\) with
and
Hence,
Finally, substituting b from (25) into (26), shows that equality is attained in the second inequality in Theorem 1 when \((5+2\lambda )\beta >2\). \(\square \)
4 \(H_2(2)(f^{-1})\) for strongly Ozaki functions
Since each class \({{\mathcal {F}}}_{O}(\lambda ,\beta )\) is compact and every \(f\in {{\mathcal {F}}}_{O}(\lambda ,\beta )\) is invertible, there exists \(r_0\in (0,1)\) such that \({\mathbb {D}}_{r_0}\subset f({\mathbb {D}})\) for every \(f\in {{\mathcal {F}}}_{O}(\lambda ,\beta ),\) where \({\mathbb {D}}_{r_0}:=\{z\in {\mathbb {C}}:|z|<r_0\}.\) Therefore, for \(w\in {\mathbb {D}}_{r_0}\) the inverse function \(f^{-1}\) can be written as
Moreover, when \(f\in {{\mathcal {F}}}_{O}(\lambda ,\beta )\) is of the form (1), then the following relations hold (see e.g., [8])
The following proposition will be used in our proof.
Proposition 1
Let \(t\in (2,11]\) and \(u\in (1,3].\) Define \(H:[0,4]\rightarrow {\mathbb {R}}\) by
where
Then, H is convex on [0, 4].
Proof
By differentiating H twice, we obtain
where
We show that our assertion is true by proving that \(G(x) \ge 0\) for \(x\in [0,4]\).
Let \(x\in [0,4]\), \(u\in (1,3]\) be fixed, and
where
and
I. Consider first the case \(A_2 \le 0\). Then, we note that
and so clearly \(F(4)>0\).
Also when \(s=121\),
where
Since \(k_3>0\) and \(1<u\le 3\), we have
and
Thus, when \(k_1 \ge 0\), it is clear that \(F(121)>0\).
If \(k_1<0\), then since \(3k_1 < k_1u\) we have
Now that
which, from (30), (31) and (32), implies that \(F(121)>0\). Thus, since \(A_2 \le 0\),
II. Next we consider the case \(A_2>0\). In this case
First note that
Also when \(s=121\), we have
where
Since \(q_3 >0\) and \(u>1\), we obtain
and further since \(q_0>0\), \(q_1>0\), and \(q_2+q_3>0\), it follows that \(\tilde{F}(121) > 0\). Hence, since the function \(\tilde{F}\) is linear with respect to s, \(\tilde{F}(4)>0\) and \(\tilde{F}(121)>0\), we obtain
and so from (33) it follows that \(F(s)>0\) for \(s\in [4,121]\).
Finally, note that I and II implied that \(F(s)>0\) for \(s\in (4,121]\), \(x\in [0,4]\), and \(u\in (1,3]\), which therefore shows that \(G(x) \ge 0\), and the proof of Proposition 1 is complete. \(\square \)
Theorem 2
Let \(\beta \in (0,1]\) and \(\lambda \in [1/2,1]\). If \(f\in {{\mathcal {F}}}_{O}(\lambda ,\beta )\), then
Both inequalities are sharp.
Proof
Note first that since \(H_{2}(2)\) and the class \( {{\mathcal {F}}}_{O}(\lambda ,\beta )\) are rotationally invariant, by (28) together with (11) and (6) we can write
where
for some \(c\in [0,2]\) and \(\zeta \), \(\eta \in \overline{{\mathbb {D}}}\).
I. Assume first that \((1+10\lambda )\beta \le 2\). Then, applying the triangle inequality to (36), and since \(|\zeta | \le 1\), we obtain
Since \(-1 -3\beta +8\lambda ^2 \beta -2\lambda (5+\beta ) < 0,\) and \(-2 + (1+10\lambda )\beta \le 0\), by (37), it follows that
and so from (35), the first inequality in (34) is proved.
II. Next assume that \((1+10\lambda )\beta > 2\). We consider two cases.
II(A) When \(\beta =1/2\) and \(\lambda =1\), from (36), we obtain
Therefore, since \(|\zeta | \le 1\) and \(c \in [0,2]\), we have
and so from (35), we obtain inequality (34).
II(B) Now assume that \(\beta \not =1/2\) or \(\lambda \not =1,\) so that \(-2 < (-3-2\lambda +8\lambda ^2)\beta ^2 \le 3\).
When \(c=0\), (36), gives
and when \(c=2\), we have
We suppose therefore that \(c\in (0,2)\) and use Lemma 2. Applying the triangle inequality to (36), we obtain
where for \(\zeta \in \overline{{\mathbb {D}}},\)
with
Clearly, \(A<0\), \(B>0\) and \(C>0\), and so \(AC<0\). Furthermore, the following conditions hold for all \(\lambda \) and \(\beta \) such that \((1+10\lambda )\beta >2\):
-
(i)
\(B^2 > -4AC( C^{-2} -1)\),
-
(ii)
\(2(1-|C|) < |B|\),
-
(iii)
\(|C|( |B|+4|A| ) > |AB|\).
Indeed (i) follows from
and (ii) written as \(B-2(1-C)>0\) is true, since
Finally, note that since \(B-4C<0\),
and \(BC>0\), so condition (iii) written as \(BC +A(B-4C) >0\) is true.
Next note that the condition \(|AB| \le |C|(|B|-4|A|)\) is equivalent to \(BC+4AC+AB \ge 0\), and a computation gives
where for \(x\in [0,4],\)
with
We note that \(b_0>0\), \(b_1<0\) and \(b_2<0\). Furthermore, \(\varphi '(x)=2b_1+2b_2x<0\) for \(x\in (0,4)\). Clearly, \(\varphi (0)=b_0>0\), and \(\varphi (4)=16b_2 +8b_1 +b_0 <0\). Indeed, by putting \(t=(1+10\lambda )\beta \) and \(u=(-3-2\lambda +8\lambda ^2)\beta ^2\), since \(t>2\) and \(u>-2\), we have
Therefore, the function \(\varphi \) has a unique zero \(x_0\in (0,4)\), where
Thus,
and it follows that
II(B)-a. Let \(c\in (0,\sqrt{x_0}]\). Then, by Lemma 2,
where for \(x\in [0,4],\)
with
We note that \(g'(x)=0\) only when \(x=\tilde{\tau }:=-d_1/d_2\), and clearly, \(\tilde{\tau }>0\).
We now show that \(\tilde{\tau } < x_0\). Since
by (41), the inequality \(\tilde{\tau } < x_0\) is equivalent to
Moreover, by putting \(t=(1+10\lambda )\beta \), and \(u=(-3-2\lambda +8\lambda ^2)\beta ^2\), we obtain
Since \(t>2\) and \(-2<u\le 3\), the quantities \(16(52+9u-4u^2)\), \(32(7-u)\) and \(4(8-u)\) are all positive. Thus, we obtain
which, by (45), implies that (44) holds. So \(\tilde{\tau }<x_0,\) and since \(d_2<0\), the function g has its maximum at \(\tilde{\tau }\) in \((0,x_0]\). Thus, by (42), we have
II(B)-b. Let \(c\in [\sqrt{x_0},2)\). Then, by Lemma 2, we obtain
where for \(x\in [0,4],\)
and
It is easy to see that \(h_2\) is decreasing in \([x_0,2)\).
(i) Moreover, when \((8\lambda ^2-3-2\lambda )\beta ^2 \le 1\), \(h_1\) is also decreasing in \([x_0,2)\). Therefore, from (47), we obtain
where g is the function defined by (43).
(ii) When \((8\lambda ^2-3-2\lambda )\beta ^2 > 1\), by putting \(t=(1+10\lambda )\beta \) and \(u=(8\lambda ^2-3-2\lambda )\beta ^2\) so that \(2<t\le 11\) and \(1<u\le 3\), we obtain
where H is the function defined by (29). By Proposition 1, we deduce that
and so by (47), (49) and (50), we obtain the inequality (48) once more.
Thus, from (38), (39), (46) and (48), inequality (46) holds for all \(c\in [0,2]\) and \(\zeta \), \(\eta \in \overline{{\mathbb {D}}}\), and so from (35), the second inequality in (34) is proved.
In order to show that the inequalities are sharp, first let \(f_1\) be defined for \(z\in {\mathbb {D}}\) by
where \(p_1(z) := (1+z^2)/(1-z^2)\) for \(z\in {\mathbb {D}}.\) Then, \(f_1 \in {{\mathcal {F}}}_{O}(\lambda ,\beta )\) with
for \(z\in {\mathbb {D}}.\) Hence, \(f_1^{-1}\) is given by
for \(w\in {\mathbb {D}}_{r_0}.\) Thus, \(H_{2,2}(f_1^{-1}) = -\beta ^2(1+2\lambda )^2/36\), which shows that the first bound in (34) is sharp when \((1+10\lambda )\beta \le 2\).
Next let \(f_2\) be defined, for \(z\in {\mathbb {D}}\) by
where
with
Since \(b \in (0,2)\), it follows that \(p_2 \in {{\mathcal {P}}}\). Therefore, \(f_2 \in {{\mathcal {F}}}_{O}(\lambda ,\beta )\) with
and
A simple calculation gives
and substituting for b in (51) into (52) we obtain
This shows that the bound in (34) is sharp for the case \((1+10\lambda )\beta > 2\). \(\square \)
5 Strongly convex functions \((\lambda =1/2)\)
When \(\lambda =1/2,\) i.e. when \(f\in {\mathcal {C}}^{\beta },\) the class of convex functions of order \(\beta \), it was shown in [9] that the bounds for \( |H_2(2)(f)|\) and \(|H_2(2)(f^{-1})|\) were the same, reflecting other invariant properties concerning the coefficients of f and \(f^{-1}\). It was shown that when \(f\in {\mathcal {C}}^{\beta },\)
Although these bounds are correct, they are not (as claimed), best possible, since \(\lambda =1/2\) in Theorems 1 and 2 give the following sharp bounds.
Theorem 3
If \(f\in {\mathcal {C}}^\beta \), then for \(0<\beta \le 1,\)
When \(\beta =1\), we deduce the following [4].
Corollary 1
If \(f\in {\mathcal {C}}\), then
We note that the proof of the weaker result above used Lemma 1, which alone was not strong enough to give the sharp estimate, whereas the additional use of Lemma 2 produces the correct estimates given in Theorem 3.
Remark 1
We finally note that when \(\lambda =1\), Theorems 1 and 2 show that there is no invariance between \(|H_{2}(2)(f)|\) and \(|H_2(2)(f^{-1})| \); however, a simple calculation gives the following strange invariance property when \(\beta =1\).
Corollary 2
If \(f\in {\mathcal {F}}_{O}(\lambda )\) with \(\lambda =\dfrac{1}{18}(8 + \sqrt{73})\), then
Both inequalities are sharp.
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Acknowledgements
The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP; Ministry of Science, ICT & Future Planning) (No. NRF-2017R1C1B5076778).
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Sim, Y.J., Lecko, A. & Thomas, D.K. The second Hankel determinant for strongly convex and Ozaki close-to-convex functions. Annali di Matematica 200, 2515–2533 (2021). https://doi.org/10.1007/s10231-021-01089-3
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DOI: https://doi.org/10.1007/s10231-021-01089-3