Abstract
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 {mathcal {P}}. 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.
Highlights
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 1⁄4 f 0ð0Þ À 1 1⁄4 0
A new way leading to results on coefficient functionals has been pointed out
The object of our study is the class of starlike functions with respect to symmetric points associated with the exponential function
Summary
Let B0 be the class of Schwarz functions, i.e., analytic functions x : D ! The main tool used to obtain those results was a lemma proved by Libera and Złotkiewicz. It is easy to predict the exact estimate of the functional and make the appropriate computations It is the case for the class SÃSðezÞ. We need the following lemmas for Schwarz functions. Lemma 4 ([2]) Let xðzÞ 1⁄4 c1z þ c2z2 þ Á Á Á be a Schwarz function and k 2 C. C4 þ ð1 þ kÞc1c3 þ c22 þ ð1 þ 2kÞc21c2 þ kc maxf; jkjg ð1:4Þ and
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