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For γ ∈ ℂ such that |γ| < π/2 and 0 ≤ β < 1, let ${}_{\gamma ,\beta }$ denote the class of all analytic functions P in the unit disk with P(0) = 1 and $Re\left(e^{i\gamma }P\left(z\right)\right)>\beta cos\gamma$ in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability $V_{}(z₀,\lambda )$ for ${\int }_0^{z₀}P\left(\zeta \right)d\zeta$ when P ranges over the class $\left(\lambda \right)=P{\in }_{\gamma ,\beta }:P^{\text{'}}\left(0\right)=2(1-\beta )\lambda e^{-i\gamma }cos\gamma .$As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.

We consider the class ${\mathscr{H}}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $\left|z\right|<1$ and normalized so that $h\left(0\right)=0=h^{\text{'}}\left(0\right)-1$ and $g\left(0\right)=0=g^{\text{'}}\left(0\right)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes ${𝒫}_H^0\left(\alpha \right)$ and ${𝒢}_H^0\left(\beta \right)$ of functions from ${\mathscr{H}}_0$ and show that if $f\in {𝒫}_H^0\left(\alpha \right)$ and $F\in {𝒢}_H^0\left(\beta \right)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha$ and $\beta$ are satisfied. In the second part we study the harmonic sections (partial...