A new class of analytic functions with positive coefficients
The authors define a family \(P_\lambda(\beta)\) \((0\leq\lambda<1\), \(1<\beta\leq 4/3)\) of analytic functions of the form \(f(z)=z+ \sum^\infty_{n=2} a_nz^n\) \((a_n\geq 0)\) in the open unit disc \(D\), satisfying the following condition \[ \text{Re}\left\{{z\bigl(\Omega^\lambda f(z)\bigr)'\over \Omega^\lambda f(z)}\right\} <\beta\;(z\in D) \] where \(\Omega^\lambda f(z)= \Gamma(2-\lambda)D^\lambda_zf(z)\) and \(D_z^\lambda f(z)={1\over\Gamma(1-\lambda)} {d\over dz}\int^z_0 {f(\zeta)d\zeta \over(z-\zeta)^{1-\lambda}}\). The authors obtain coefficient estimates, distortion theorem and the effect of taking Hadamard product of members of \(P_\lambda(\beta)\) for different parameters \(\lambda\) and \(\beta\). The results are simple consequences of known theorems. The wrong definition of convexity and an unknown subscript \(k\) in the definition of \(P_k(\alpha)\) which confuses the definition of \(P_\lambda(\beta)\) in the introduction could have been avoided.
scientific article; zbMATH DE number 1942788