On certain subclasses of analytic functions associated with hypergeometric functions
The authors introduce the new class of analytic functions \(TS_p(\lambda,\alpha,\beta)=S_p(\lambda,\alpha,\beta)\cap T\) where \(S_p(\lambda,\alpha,\beta)\) and \(T\) are the subclasses of \[ \mathcal A=\Big\{f \text{ holomorphic in the unit disc } U: f(z)=z+\sum_{n=2}^\infty a_nz^n\Big\} \] defined by the properties \[ \mathrm{Re}\left\{\frac{zf'(z)}{(1-\lambda)f(z)+\lambda zf'(z)}-\alpha\right\}> \beta\left|\frac{zf'(z)}{(1-\lambda)f(z)+\lambda zf'(z)}-1\right|,\quad z\in U, \] and \[ f(z)=z-\sum_{n=2}^\infty |a_n|z^n, \] respectively. As noted in the paper, the class \(TS_p(\lambda,\alpha,\beta)\) is a generalization of many other known classes of analytic functions. The main result gives necessary and sufficient conditions on the parameters \(a,b,c,\lambda, \alpha, \beta \) for functions of the form \(zF(a,b;c;z)\) and \(z[2-F(a,b;c;z)]\) to be in \(TS_p(\lambda,\alpha,\beta)\), where \[ F(a,b;c;z)=z-\left|\frac{ab}{c}\right|\sum_{n=2}^\infty \frac{(a+1)_{n-2}(b+1)_{n-2}}{(c+1)_{n-2}(1)_{n-1}}z^n. \]
scientific article; zbMATH DE number 5839402
scientific article