# Turkish Journal of Mathematics

## DOI

10.3906/mat-1808-133

## Abstract

Let $(\beta_n)_{n\ge 2}$ be a sequence of nonnegative real numbers and $\delta$ be a positive real number. We introduce the subclass $\mathcal{A}(\beta_n,\delta)$ of analytic functions, with the property that the Taylor coefficients of the function $f$ satisfies $\sum_{n\ge2}^{\infty}\beta_n a_n \le \delta$, where $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. The class $\mathcal{A}(\beta_n,\delta)$ contains nonunivalent functions for some choices of $(\beta_n)_{n\ge 2}$. In this paper, we provide some general properties of functions belonging to the class $\mathcal{A}(\beta_n,\delta)$, such as the radii of univalence, distortion theorem, and invariant property. Furthermore, we derive the best approximation of an analytic function in such class by using the semiinfinite quadratic programming. Applying our results, we recover some known results on subclasses related to coefficient inequality. Some applications to starlike and convex functions of order $\alpha$ are also mentioned.

## Keywords

Analytic function, starlike function of order $\alpha$, convex function of order $\alpha$, coefficient inequality, quadratic programming, Karush-Kuhn-Tucker conditions

## First Page

630

## Last Page

647

## Recommended Citation

WONGSAIJAI, BEN and Sukantamala, Nattakorn
(2019)
"Properties of a generalized class of analytic functions with coefficient inequality,"
*Turkish Journal of Mathematics*: Vol. 43:
No.
2, Article 6.
https://doi.org/10.3906/mat-1808-133

Available at:
https://journals.tubitak.gov.tr/math/vol43/iss2/6