Duality for convolution on subclasses of analytic functions and weighted integral operators

Abstract

Abstract In this article, we investigate a class of analytic functions defined on the unit open disc U = { z : ∣ z ∣ < 1 } {\mathcal{U}}=\left\{z:| z| \lt 1\right\} , such that for every f ∈ P α ( β , γ ) f\in {{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma ) , α > 0 \alpha \gt 0 , 0 ≤ β ≤ 1 0\le \beta \le 1 , 0 < γ ≤ 1 0\lt \gamma \le 1 , and ∣ z ∣ < 1 | z| \lt 1 , the inequality Re f ′ ( z ) + 1 − γ α γ z f ″ ( z ) − β 1 − β > 0 {\rm{Re}}\left\{\frac{f^{\prime} \left(z)+\frac{1-\gamma }{\alpha \gamma }z{f}^{^{\prime\prime} }\left(z)-\beta }{1-\beta }\right\}\gt 0 holds. We find conditions on the numbers α , β \alpha ,\beta , and γ \gamma such that P α ( β , γ ) ⊆ S P ( λ ) {{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma )\subseteq SP\left(\lambda ) , for λ ∈ ( − π 2 , π 2 ) \lambda \in \left(-\frac{\pi }{2},\frac{\pi }{2}) , where S P ( λ ) SP\left(\lambda ) denotes the set of all λ \lambda -spirallike functions. We also make use of Ruscheweyh’s duality theory to derive conditions on the numbers α , β , γ \alpha ,\beta ,\gamma and the real-valued function φ \varphi so that the integral operator V φ ( f ) {V}_{\varphi }(f) maps the set P α ( β , γ ) {{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma ) into the set S P ( λ ) SP\left(\lambda ) , provided φ \varphi is non-negative normalized function ( ∫ 0 1 φ ( t ) d t = 1 ) \left({\int }_{0}^{1}\varphi \left(t){\rm{d}}t=1) and V φ ( f ) ( z ) = ∫ 0 1 φ ( t ) f ( t z ) t d t . {V}_{\varphi }(f)\left(z)=\underset{0}{\overset{1}{\int }}\varphi \left(t)\frac{f\left(tz)}{t}{\rm{d}}t.