A Class of Analytic Functions with Missing Coefficients

Abstract

Let $T_n(A,B,\gamma ,\alpha )(-1\le B<1,B<A,0<\gamma \le 1$ and $\alpha >0)$ denote the class of functions of the form $f\left(z\right)=z+{{\displaystyle \sum }}_{k=n+1}^{\mathrm{\infty }}{a}_k{z}^{\mathrm{k}}(n\in N=\{1,2,3,\dots \}),$ which are analytic in the open unit disk $U$ and satisfy the following subordination condition $f^{\prime }\left(z\right)+\alpha z{f}^{\prime \prime }\left(z\right)\prec {\left(\right(1+Az)/(1+Bz\left)\right)}^{\gamma }$ , for $(z\in U;A\le 1;0<\gamma <1),(1+Az)/(1+Bz)$, for $(z\in U;\gamma =1)$. We obtain sharp bounds on $\text{R}\text{e}f\text{'}\left(z\right),\text{R}\text{e}f\left(z\right)/z,\left|f\left(z\right)\right|$ , and coefficient estimates for functions $f\left(z\right)$ 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.