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On the logarithmic coefficients of Ma-Minda type convex functions

Md Firoz Ali, Lokenath Thakur

Abstract

In this paper, we investigate three specific subclasses of Ma-Minda type convex functions: namely, convex functions of order $α$, Janowski convex functions, and Robertson functions of normalized analytic functions defined in the open unit disk. For these classes, we establish logarithmic coefficient inequalities concerning both individual coefficient estimates and weighted series. The results presented here correct some earlier erroneous results and extend several previously known ones.

On the logarithmic coefficients of Ma-Minda type convex functions

Abstract

In this paper, we investigate three specific subclasses of Ma-Minda type convex functions: namely, convex functions of order , Janowski convex functions, and Robertson functions of normalized analytic functions defined in the open unit disk. For these classes, we establish logarithmic coefficient inequalities concerning both individual coefficient estimates and weighted series. The results presented here correct some earlier erroneous results and extend several previously known ones.
Paper Structure (7 sections, 15 theorems, 106 equations, 1 table)

This paper contains 7 sections, 15 theorems, 106 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. The class $\mathcal{F}(c)$
  4. The class $\mathcal{C}(A,B)$
  5. The class $\mathcal{S}_\alpha$
  6. Auxiliary Results
  7. Proof of the main results

Key Result

Theorem A

cho-Alimohammadi-2021 For $f\in \mathcal{F}(c)$ with $c \in (0, 0.656] \cup \{2\}$, the logarithmic coefficients of $f$ satisfies the inequality and where $D_n$ are the Taylor coefficients of $\psi(z)$ given by T-fun-002. Further, both the inequalities are sharp.

Theorems & Definitions (19)

  • Theorem A
  • Proposition 1
  • Theorem B
  • Theorem 2.1
  • Theorem C
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • ...and 9 more