Initial logarithmic coefficients for functions starlike with respect to symmetric points

In this paper, we obtain the bounds of the initial logarithmic coefficients for functions in the classes SS∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_S^*$$\end{document} and KS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}_S$$\end{document} of functions which are starlike with respect to symmetric points and convex with respect to symmetric points, respectively. In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in SS∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_S^*$$\end{document} and KS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}_S$$\end{document} by the corresponding coeffcients of Schwarz functions. In the proofs, we apply some inequalities for these functions obtained by Prokhorov and Szynal, by Carlson and by Efraimidis. This approach offers a additional benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for SS∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_S^*$$\end{document} and KS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}_S$$\end{document}.


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 : : It is known (see, [7]) that K & S Ã S and S Ã ð2Þ & S Ã S , where S Ã ð2Þ stands for the class of odd starlike functions. On the other hand, S Ã S & C. Our aim is to derive the bounds of the initial logarithmic coefficients for functions in the both classes defined above. It is worth observing that the Koebe function does not belong to S Ã S , so it cannot play a role of extremal function for S Ã S . In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in S Ã S and K S by the corresponding coeffcients of Schwarz functions. This makes the calculation easier. Additionally, this approach offers a valuable benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for S Ã S and K S . Let B 0 be the class of Schwarz functions, i.e., analytic functions x : D ! D, xð0Þ ¼ 0. A function x 2 B 0 can be written as a power series To prove our results, we need the following lemmas for Schwarz functions obtained by Prokhorov and Szynal [6], by Carlson ([1]) and by Efraimidis [2].
Then jc 2 j 1 À jc 1 j 2 ; jc 3 j 1 À jc 1 j 2 À jc 2 j 2 1 þ jc 1 j ; jc 4 j 1 À jc 1 j 2 À jc 2 j 2 : Lemma 3 is a particular case of more general theorem which is needed in our proofs. For p in P, the class of analytic functions p such that RepðzÞ [ 0 and pð0Þ ¼ 1, and for w 2 C, Efraimidis in [2]  and proved the following theorem.
Theorem 1 If p 2 P and w 2 C, then jA k;n ðwÞj 2 maxf1; j1 À 2wj k g for all integers k ! 0 and n ! 1.
Applying the correspondence between p 2 P and x 2 B 0 , it is possible to obtain the analogous theorem for Schwarz functions. As a corollary, putting w ¼ 0 and k ¼ 3, n ¼ 1, Lemma 3 follows. Consider now the case k þ n ¼ 5 in

ð1:9Þ
Combining Formulae (1.7-1.9) for suitably chosen l we can get other relative inequalities; some of them will be useful in proving theorems from the two next sections.
Taking l ¼ 0 and l ¼ À1 in (1.9), we have From (1.7) with l ¼ 1 and l ¼ À1, it follows that Finally, (1.11) and (1.12) results in 2 Logarithmic coefficients for functions in S *

S
Although the first three results in the theorem below are easy to obtain in other way, for completeness of results, we prove all of them in a uniform way.
To derive the bound of c 4 , we can write By Lemma 3, the first expression in square brackets is bounded by 1. Now, it is enough to apply Lemma 2 together with the triangle inequality to obtain In this way, jc 4 j 1 4 . The coefficient c 5 can be rewritten as follows The first component in the square brackets is bounded by 1. It is a simple consequence of Formula (1.7) with l ¼ 1.
The second component is equal to W 1 þ W 2 , where :
The function H as a quadratic function of a variable y achieves its greatest value when y ¼ 5 8 z. Consequently, Hðx; y; zÞ Hðx; 5 8 z; zÞ ¼ 1 À x 2 þ 25 64 Since gðx; zÞ ! 0 for x 2 ½0; 1 and z 2 ½0; 1, so Hðx; y; zÞ 1 : ð2:6Þ Taking into account (2.4-2.6), and consequently, : For the sharpness of the results, it is enough to observe that taking xðzÞ ¼ z k leads to equalities in each inequality in Theorem 2. h It is worth to write the extremal functions f 2 S Ã S and the corresponding functions explicitly. This is easy for even logarithmic coefficients. One can check that Indeed, this function is generated from the odd starlike function f ðzÞ ¼ z 1Àz 2 under the n-th root transformation. Since this transformation preserves starlikeness, we can see that f 2n given by (2.8

) is in S Ã ð2Þ
. Hence, it is in S Ã S . For f 2n , we derive For n ¼ 1, the extremal function is f 1 ðzÞ ¼ z 1Àz ; for this function Consider now other positive odd integers and define a function : ð2:9Þ Observe that f 2nþ1 is a difference of the two components. Denote them by g and h, respectively. Clearly, 1 2 gðzÞ À gðÀzÞ p and 1 2 hðzÞ À hðÀzÞ ½ ¼0 : ð2:10Þ Moreover, On the other hand, since ð2:12Þ Combining (2.10-2.12) leads to 2zf 0 2nþ1 ðzÞ f 2nþ1 ðzÞ À f 2nþ1 ðÀzÞ ¼ 2zg 0 ðzÞ À 2zh 0 ðzÞ gðzÞ À gðÀzÞ This means that f 2nþ1 belongs to the class S Ã S . From (2.9), we conclude that f 2nþ1 is ð2n þ 1Þ-fold symmetric function, i.e. f 2nþ1 ðezÞ ¼ ef 2nþ1 ðzÞ with e ¼ expð2p=ð2n þ 1ÞÞ being a root of order 2n þ 1 of unity, and Hence, the corresponding function F 2nþ1 is also ð2n þ 1Þ-fold symmetric and The above proves that the third logarithmic coefficient is equal to 1/4 for f 3 and the fifth logarithmic coefficient is equal to 1/6 for f 5 . The natural conjecture is following Applying the inequalities from Lemma 2 and writing x ¼ jc 1 j, y ¼ jc 2 j, x; y 2 ½0; 1, the expression V can be estimated as follows
Author Contributions Not applicable. Funding The project/research was financed in the framework of the project Lublin University of Technology-Regional Excellence Initiative, funded by the Polish Ministry of Science and Higher Education (contract no. 030/RID/2018/19).
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Declarations
Conflict of interest The authors declare that they have no conflict of interest.
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