Some functional inequalities and inclusion relationships associated with certain families of integral operators
Several subclasses of starlike, convex, close-to-convex and quasi-convex functions are considered. Let \(A\) be the class of functions \(f\) normalized by \[ f(z)=z+\sum^\infty_{n=2}a_nz^n, \] which are analytic in the open unit disk \(U:=\{z\in\mathbb{C}\mid|z|<1\}\). Let \(\alpha,\beta\) be real numbers with \(0\leq \alpha\), \(\beta<1\). Let \[ \begin{aligned} {\mathcal S}^* (\alpha): & =\left\{f\in A\mid \text{Re}\left(\frac{zf'(z)}{f(z)}\right)> \alpha,\quad\forall z\in U\right\},\\ \mathbb{K} (\alpha): & =\left\{f\in A\mid \text{Re}\left(1+\frac {zf''(z)}{f'(z)} \right)> \alpha,\quad\forall z\in U \right\},\\ {\mathcal C}(\beta,\alpha): & =\left\{ f\in A\mid g \in {\mathcal S}^*(\alpha)\text{ and }\text{Re}\left(\frac {zf'(z)}{g(z)}\right)> \beta, \quad \forall z\in U\right\},\\ {\mathcal C}^*(\beta,\alpha): & =\left\{f\in A \mid g\in\mathbb{K} (\alpha)\text{ and }\text{Re}\left(\frac{zf''(z)+f'(z)} {g'(z)} \right)> \beta,\quad\forall z\in U\right\},\end{aligned} \] and let \[ Q_\lambda^\mu f(z):=z+ \frac{\Gamma(\lambda+\mu+1)} {\Gamma(\lambda+1)} \sum^\infty_{n=2}\frac {\Gamma (\lambda+n)}{\Gamma(\lambda+ \mu+n)}a_nz^n \] for \(\lambda>-1\), \(\mu>0\), \(f\in A\), where \(\Gamma\) denotes the Gamma function. Let \[ \begin{aligned} {\mathcal S}^*_{\lambda, \mu} (\alpha): & =\bigl\{f\in A\mid Q_\lambda^\mu \mid Q^\mu_\lambda f\in{\mathcal S}^* (\alpha)\bigr\},\\ \mathbb{K}_{\lambda, \mu}(\alpha): & =\bigl\{f\in A\mid Q_\lambda^\mu f\in \mathbb{K}(\alpha) \bigr\},\\ {\mathcal C}_{\lambda,\mu}(\beta,\alpha): & =\bigl\{ f\in A\mid Q_\lambda^\mu f\in{\mathcal C}(\beta,\alpha)\bigr\} \end{aligned} \] and \[ {\mathcal C}^*_{\lambda,\mu}(\beta,\alpha):=\bigl\{f\in A\mid Q_\lambda^\mu f \in{\mathcal C}^*(\beta,\alpha)\bigr\}. \] Using the Miller-Mocanu lemma [\textit{S. S. Miller} and \textit{P. T. Mocanu}, J. Math. Anal. Appl. 65, 289--305 (1978; Zbl 0367.34005)], the authors prove that the inclusions \({\mathcal S}^*_{\lambda,\mu} (\alpha)\subset{\mathcal S}^*_{\lambda,\mu+1}(\alpha)\), \(\mathbb{K}_{\lambda,\mu}(\alpha) \subset\mathbb{K}_{\lambda,\mu+1}(\alpha)\), \({\mathcal C}_{\lambda,\mu} (\beta,\alpha) \subset{\mathcal C}_{\lambda,\mu+1}(\beta,\alpha)\) and \({\mathcal C}^*_{\lambda,\mu} (\beta,\alpha)\subset{\mathcal C}^*_{\lambda, \mu+1} (\beta,\alpha)\) hold.
scientific article; zbMATH DE number 5003100