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### Convolution theorems for starlike and convex functions in the unit disc

Annales Polonici Mathematici

Let A denote the space of all analytic functions in the unit disc Δ with the normalization f(0) = f’(0) − 1 = 0. For β < 1, let $P{⁰}_{\beta }=f\in A:Re{f}^{\text{'}}\left(z\right)>\beta ,z\in \Delta$. For λ > 0, suppose that denotes any one of the following classes of functions: ${M}_{1,\lambda }^{\left(1\right)}=f\in :Rez{\left(z{f}^{\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}>-\lambda ,z\in \Delta$, ${M}_{1,\lambda }^{\left(2\right)}=f\in :Rez{\left(z²{f}^{\text{'}\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}>-\lambda ,z\in \Delta$, ${M}_{1,\lambda }^{\left(3\right)}=f\in :Re1/2{\left(z{\left(z²{f}^{\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}\right)}^{\text{'}}-1>-\lambda ,z\in \Delta$. The main purpose of this paper is to find conditions on λ and γ so that each f ∈ is in ${}_{\gamma }$ or ${}_{\gamma }$, γ ∈ [0,1/2]. Here ${}_{\gamma }$ and ${}_{\gamma }$ respectively denote the class of all starlike functions of order γ and the class of all convex functions of order γ. As a consequence, we obtain a number...

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