On coefficient problems for functions starlike with respect to symmetric points

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 $${\mathcal {P}}$$ 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.

Hankel Determinant for a Subclass of Starlike Functions with Respect to Symmetric Points Subordinate to the Exponential Function

“…s (e z ), then H 3,1 = 13 128 . Comparing references [31,33], it is evident that Zaprawa's research results improved upon some of Ganesh's conclusions.…”

“…In this paper, upper-bound estimates for the Hankel determinant of the class S * S (e z ) are studied using a new Lemma and method. Theorem 5 in reference [33] is proved once again, and the extremal function is obtained. The upper-bound estimate of |H 3,1 ( f )| for the functions class S * s (e z ) is more accurate than Theorem 6 in reference [33].…”

“…Theorem 5 in reference [33] is proved once again, and the extremal function is obtained. The upper-bound estimate of |H 3,1 ( f )| for the functions class S * s (e z ) is more accurate than Theorem 6 in reference [33]. The proof of Theorem 2 indicates that the exact upper-bound estimate of |H 3,1 ( f )| cannot be obtained at the boundary of Ω.…”

“…This method is based on the use of a lemma derived by Carlson [34] about the coefficients of the Schwarz function. In particular, for the Hankel determinant H 3,1 , we obtained a more precise upper-bound estimate than in [33].…”

“…In 2022, Zaprawa [33] researched the class S * s (e z ) using a new approach, which is based on relating the coefficients of functions in a given class with the coefficients of corresponding Schwarz functions. Some coefficient estimates of the function class S * s (e z ) were obtained, including logarithmic coefficient estimates, Zalcman functional estimates and upper bounds for the second-and third-order Hankel determinants, etc.…”