On the subordination and superordination of strongly starlike functions (Q2836161)

Using some classical subordination and superordination results, for a function \(p\) analytic in the open unit disk \(\mathbb {U}\) with \(p(0)=1\), the authors gave a sufficient condition that implies \(\left | \arg p(z)\right | <\pi \beta /2\). Also, they prove a general result that represents a sufficient condition for a function to be in the some subclass of strongly starlike functions. Moreover, the authors obtained a sharp sandwich-type result, where they found sufficient conditions on \(f\) and \(g\) such that NEWLINE\[NEWLINE q_{1}^{2}(z)+zq_{1}^{\prime}(z)\prec \frac {zf^{\prime}(z)}{g(z)}\Biggl (1+\frac {zf^{\prime \prime}(z)}{f^{\prime}(z)}-\frac {zg^{\prime}(z)}{g(z)}+\frac {zf^{\prime}(z)}{g(z)}\Biggr)\prec q_{2}^{2}(z)+zq_{2}^{\prime}(z) NEWLINE\]NEWLINE implies NEWLINE\[NEWLINE q_{1}(z)\prec \frac {zf^{\prime}(z)}{g(z)}\prec q_{2}(z), NEWLINE\]NEWLINE where \(q_{1}\) and \(q_{2}\) are two convex functions with positive real part in \(\mathbb {U}\).