## Abstract and Applied Analysis

### A Class of Analytic Functions with Missing Coefficients

#### Abstract

Let ${T}_{n}(A,B,\gamma ,\alpha )(-1\le B\lt 1,B\lt A,0\lt \gamma \le 1$ and $\alpha >0)$ denote the class of functions of the form $f(z)=z+{\sum }_{k=n+1}^{\mathrm{\infty }}{a}_{k}{z}^{\mathrm{k}}(n\in N=\{1,2,3,\dots \}),$ which are analytic in the open unit disk $U$ and satisfy the following subordination condition ${f}^{\prime }(z)+\alpha z{f}^{\prime \prime }(z)\prec {((1+Az)/(1+Bz))}^{\gamma }$ , for $(z\in U;A\le 1;0\lt \gamma \lt 1),(1+Az)/(1+Bz)$, for $(z\in U;\gamma =1)$. We obtain sharp bounds on $\text{R}\text{e}f'(z),\text{R}\text{e}f(z)/z,|f(z)|$ , and coefficient estimates for functions $f(z)$ belonging to the class ${T}_{n}(A,B,\gamma ,\alpha )$. Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.

#### Article information

Source
Abstr. Appl. Anal., Volume 2011 (2011), Article ID 456729, 16 pages.

Dates
First available in Project Euclid: 12 August 2011

https://projecteuclid.org/euclid.aaa/1313171208

Digital Object Identifier
doi:10.1155/2011/456729

Mathematical Reviews number (MathSciNet)
MR2576838

Zentralblatt MATH identifier
1220.30028

#### Citation

Yang, Ding-Gong; Liu, Jin-Lin. A Class of Analytic Functions with Missing Coefficients. Abstr. Appl. Anal. 2011 (2011), Article ID 456729, 16 pages. doi:10.1155/2011/456729. https://projecteuclid.org/euclid.aaa/1313171208