Subclasses of uniformly starlike and convex functions defined by certain integral operator (Q882692)

Two function classes \({\mathcal T}Q(\alpha ,\beta ,\lambda )\) and \({\mathcal T}{\mathcal J}(\alpha ,\sigma )\) are introduced by defining these classes, respectively, in terms of the functions \(F(z) = Q_\beta ^\alpha f(z)\) and \(F(z) = J_\alpha f(z),\) involving certain integral operators and satisfying the inequality: \[ \left| {\frac{{z\left( {F(Z)} \right)^\prime }}{{F(Z)}} - 1} \right| \leq \text{Re} \left\{ {\frac{{z\left( {F(Z)} \right)^\prime }}{{F(Z)}}} \right\} - \sigma \quad (z \in D), \] where \(\alpha>0, \beta>-1\: \text{and} \: 0\leq\sigma\leq1,\) D is the open unit disk, and \( f(z) = z - \sum_{n = 2}^\infty {a_n z^n }.\) The coefficient inequalities for the function \(f(z)\) belonging to the above classes (which are of uniformly starlike type) are obtained, and also similar investigations are undertaken for the functions which are of uniformly convex type.