## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 46, Number 1 (2017), 15-27.

### A generalization of starlike functions of order alpha

Sarita AGRAWAL and Swadesh Kumar SAHOO

#### Abstract

For every $q\in(0,1)$ and $0\le \alpha \lt 1$ we define a class of analytic functions, the so-called $q$-starlike functions of order $\alpha$, on the open unit disk. We study this class of functions and explore some inclusion properties with the well-known class of starlike functions of order $\alpha$. The paper is also devoted to the discussion on the Herglotz representation formula for analytic functions $zf'(z)/f(z)$ when $f(z)$ is $q$-starlike of order $\alpha$. As an application we also discuss the Bieberbach conjecture problem for the $q$-starlike functions of order $\alpha$.

#### Article information

**Source**

Hokkaido Math. J., Volume 46, Number 1 (2017), 15-27.

**Dates**

First available in Project Euclid: 30 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1498788094

**Digital Object Identifier**

doi:10.14492/hokmj/1498788094

**Mathematical Reviews number (MathSciNet)**

MR3677873

**Zentralblatt MATH identifier**

1361.30017

**Subjects**

Primary: 28A25: Integration with respect to measures and other set functions 30B10: Power series (including lacunary series) 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C50: Coefficient problems for univalent and multivalent functions 30C55: General theory of univalent and multivalent functions 33B10: Exponential and trigonometric functions 39A13: Difference equations, scaling ($q$-differences) [See also 33Dxx] 39A70: Difference operators [See also 47B39] 40A20: Convergence and divergence of infinite products 46G05: Derivatives [See also 46T20, 58C20, 58C25] 47B38: Operators on function spaces (general) 47B39: Difference operators [See also 39A70]

**Keywords**

Starlike functions q-starlike functions order of starlikeness order of q-starlikeness q-difference operator Bieberbach's conjecture infinite product uniform convergence Herglotz representation probability measure

#### Citation

AGRAWAL, Sarita; SAHOO, Swadesh Kumar. A generalization of starlike functions of order alpha. Hokkaido Math. J. 46 (2017), no. 1, 15--27. doi:10.14492/hokmj/1498788094. https://projecteuclid.org/euclid.hokmj/1498788094