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

In this article, we investigate a class of analytic functions defined on the unit open disc U = { z : ∣ z ∣ < 1 } {U}=\{z:| z| < 1\} , such that for every f ∈ P α ( β , γ ) f∈ {{P}}α(β ,γ ) , α > 0 α > 0 , 0 ≤ β ≤ 1 0≤ β ≤ 1 , 0 < γ ≤ 1 0< γ ≤ 1 , and ∣ z ∣ < 1 | z| < 1 , the inequality Re f ′ ( z ) + 1 − γ α γ z f ″ ( z ) − β 1 − β > 0 {Re}\{{f^{} (z)+1-γ /α γ z{f}^{^{} }(z)-β }{1-β }\}> 0 holds. We find conditions on the numbers α , β α ,β , and γ γ