A Class of Analytic Functions with Missing Coefficients - Ding-Gong Yang; Jin-Lin Liu (Abstract and Applied Analysis)
Let 𝑇𝑛(𝐴,𝐵,𝛾,𝛼)(−1≤𝐵<1,𝐵<𝐴,0<𝛾≤1 and 𝛼>0) denote the class of functions of the form ∑𝑓(𝑧)=𝑧+∞𝑘=𝑛+1𝑎𝑘𝑧𝑘(𝑛∈𝑁={1,2,3,…}), which are analytic in the open unit disk 𝑈 and satisfy the following subordination condition 𝑓(𝑧)+𝛼𝑧𝑓(𝑧)≺((1+𝐴𝑧)/(1+𝐵𝑧))𝛾, for(𝑧∈𝑈;𝐴≤1;0<𝛾<1),(1+𝐴𝑧)/(1+𝐵𝑧), for(𝑧∈𝑈;𝛾=1). We obtain sharp bounds on Re𝑓(𝑧),Re𝑓(𝑧)/𝑧,|𝑓(𝑧)|, and coefficient estimates for functions 𝑓(𝑧) belonging to the class 𝑇𝑛(𝐴,𝐵,𝛾,𝛼). Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.