On coefficient problems for functions starlike with respect to symmetric points

The main idea of the study on coefficient problems in various classes of analytic functions (univalent or nonunivalent) is to express the coefficients of functions in a given class by the coefficients of corresponding functions with positive real part. Thus, coefficient functionals can be studied using inequalities known for the class P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}$$\end{document}. Lemmas obtained by Libera and Złotkiewicz and by Prokhorov and Szynal play a special role in this approach. Recently, a new way leading to results on coefficient functionals has been pointed out. This approach is based on relating the coefficients of functions in a given class and the coefficients of corresponding Schwarz functions. In many cases, if we follow this approach, it is easy to predict the exact estimate of the functional and make the appropriate computations. In the proofs of these estimates are used not only classical results (the Schwarz–Pick Lemma or Wiener’s inequality), but also inequalities obtained either recently (e.g. by Efraimidis) or long ago yet almost forgotten (Carlson’s inequality). In this paper, a number of coefficient problems will be solved using the new approach described above. The object of our study is the class of starlike functions with respect to symmetric points associated with the exponential function.


Introduction
Let D be the unit disk fz 2 C : jzj\1g and A be the family of all functions f analytic in D, normalized by the condition f ð0Þ ¼ f 0 ð0Þ À 1 ¼ 0. It means that f has the expansion f ðzÞ ¼ z þ X 1 n¼2 a n z n : ð1:1Þ Let also B 0 be the class of Schwarz functions, i.e., analytic functions x : D ! D, xð0Þ ¼ 0. The function x 2 B 0 can be written as a power series where u 0 ðzÞ ¼ 1 þ z 1 À z : If the function u 0 ðzÞ is replaced by any analytic univalent function u with positive real part in D and symmetric with respect to the real axis, then we obtain the class S Ã S ðuðzÞÞ.
In this paper, we consider the class S Ã S ðuðzÞÞ with uðzÞ ¼ e z . Hence, we can write 2zf 0 ðzÞ f ðzÞ À f ðÀzÞ ¼ e xðzÞ ; x 2 B 0 ; z 2 D

& '
: This class was first discussed by Ganesh et al. in [3], where some coefficients functionals were estimated. The majority of results were not sharp. The main tool used to obtain those results was a lemma proved by Libera and Złotkiewicz.
In this paper, we follow a new approach which is based on relating the coefficients of functions in a given class and the coefficients of corresponding Schwarz functions. In many cases, it is easy to predict the exact estimate of the functional and make the appropriate computations. It is the case for the class S Ã S ðe z Þ. By applying the lemmas proved by Libera and Złotkiewicz and by Prokhorov and Szynal as well as some other tools, and by performing the calculus more precisely, we are able to derive better estimates, almost all of them being sharp.
To prove our results, we need the following lemmas for Schwarz functions. The first one is the above-mentioned result obtained by Prokhorov and Szynal.
Then, for any real numbers l and m such that ðl; mÞ 2 jlj the following sharp estimate holds From the Schwarz-Pick Lemma, it follows that for x 2 B 0 of the form (1.2), This inequality can be improved (see, for example, [2]) as follows. For any k 2 C, Carlson in [1] obtained another generalization of the Schwarz-Pick Lemma. Here, we state only these inequalities which are useful for our purpose (for all details, see [7]).

:
The above lemma immediately results in the following fact.
We also need the results obtained by Efraimidis. The method used by Efraimidis in the proof of his lemma has much greater potential. Based on this method, we can obtain some inequalities involving the fifth coefficient of x 2 B 0 (see also [6]).

Coefficient bounds
We start with the coefficients of f 2 S Ã S ðe z Þ. Applying in 2zf 0 ðzÞ f ðzÞ À f ðÀzÞ ¼ e xðzÞ ; ð2:1Þ the expansions of f and x given by (1.1) and (1.2), we obtain ð2:2Þ From [3], it is known that if f 2 S Ã S ðe z Þ is of the form (1.1), then ja 2 j 1 2 and ja 3 j 1 2 . : The bounds are sharp.
Proof Lemma 1 with l ¼ 3 2 and m ¼ 5 12 applied to results in the first inequality.
To prove the second inequality, we can write Combining the estimates of both components of (2.3), we get ja 5 j 1 4 .
This means that the equalities in the assertion of this theorem hold for the functions given by (2.1) with xðzÞ ¼ z 3 and xðzÞ ¼ z 4 , respectively. h The logarithmic coefficients of a given univalent function f, denoted by c n ¼ c n ðf Þ, are defined as If f is given by (1.1), then its logarithmic coefficients are given as follows : The bounds are sharp.

ð2:6Þ
The bounds of c 1 and c 2 are clear. The result for c 3 immediately follows from Lemma 1 with l ¼ 1 2 and m ¼ 1 12 . Observe that 1 À jc 1 j 2 À jc 2 j 2 þ 1 2 which is clearly less than or equal to 1/2. For c 5 , we have The above expression takes its greatest value with respect to jc 3 j when jc 3 j ¼ 1 4 ðjc 2 j þ jc 1 j 2 Þð1 þ jc 1 jÞ, so Combining all these inequalities, we get hðc; dÞ hð0; 0Þ ¼ 1 ; which results in the desired bound.
Observe that we obtain equalities in each bound of c k , k ¼ 1; 2; 3; 4; 5 when xðzÞ ¼ z k . This means that the obtained estimates are sharp. h

Estimates of Zalcman functionals and Hankel determinants
It is known ( [3]) that if f 2 S Ã S ðe z Þ is of the form (1.1), then ja 3 À a 2 2 j 1 2 . This functional, known as the Fekete-Szegö functional, is a particular case of the Zalcman functional a nþmÀ1 À a n a m . Let us consider other cases of the Zalcman functional.

:
Applying it and jc 4 j 1 À jc 1 j 2 À jc 2 j 2 , we get But h is a decreasing function of the variable c; consequently, hðc; dÞ hð0; dÞ The function h(0, d) achieves its greatest value in [0, 1] if d ¼ 1=2, so hð0; dÞ 13 8 , which completes the proof. h This result is not sharp. Based on Formula (3.1), it is expected that the sharp bound of jH 3;1 j is equal to 1 16 . The method used in the proof of Theorem 2 for the bound jc 5 j 1 12 may be adopted to prove that if f 2 S Ã S ðe z Þ is of the form (1.1), then ja 6 À a 2 a 5 j 1 6 . This result is also sharp. Unfortunately, the sharp bound of a 6 has not been obtained. From (2.2), we can only obtain that with an obvious conjecture that the exact value of the bound is equal to 1/6.
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