Let f be an analytic and univalent function in an open unit disk 𝔻 that belongs to certain subclasses, such as starlike, convex, or close-to-convex functions. For the parameters α, β ∈ [0, 1] such that α + β ≤ 1, we define a function<InlineEquation ID="IEq1"> <EquationSource Format="TEX">${g}_{\alpha ,\beta }\left(z\right)=\left(1-\alpha -\beta \right)f\left(z\right)+\left(\alpha +\beta \right)z{f}^{^{\prime}}\left(z\right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>=</mo> <mfenced close=")" open="("> <mn>1</mn> <mo>-</mo> <mi>α</mi> <mo>-</mo> <mi>β</mi> </mfenced> <mi>f</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>+</mo> <mfenced close=")" open="("> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mfenced> <mi>z</mi> <mmultiscripts> <mrow> <mi>f</mi> </mrow> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation>which represents a convex-type combination of the identity operator and the classical differential operator. We investigate the conditions under which the function gα,β(z) generated by a generalized linear operator preserves the geometric properties of the original function f with particular emphasis on radius problems related to univalence and distortion behavior. Explicit radius bounds are deduced by using classical analytic techniques. In addition, AI-assisted numerical experiments are used to verify the sharpness of the theoretical results and to illustrate the dependence of the radius functions on the parameters α and β. Representative numerical values and graphical visualizations are provided.

Geometric Properties of a Generalized Linear Operator Acting on Univalent Functions

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发表日期：

2026年5月29日

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Alaattin Akyar,
Oya Mert Coşkun

摘要

Let f be an analytic and univalent function in an open unit disk 𝔻 that belongs to certain subclasses, such as starlike, convex, or close-to-convex functions. For the parameters α, β ∈ [0, 1] such that α + β ≤ 1, we define a function

${g}_{\alpha ,\beta }\left(z\right)=\left(1-\alpha -\beta \right)f\left(z\right)+\left(\alpha +\beta \right)z{f}^{^{\prime}}\left(z\right),$ $g_{α, β} (z) = (1 - α - β) f (z) + (α + β) z f^{^{'}} (z),$

which represents a convex-type combination of the identity operator and the classical differential operator. We investigate the conditions under which the function g_α,_β(z) generated by a generalized linear operator preserves the geometric properties of the original function f with particular emphasis on radius problems related to univalence and distortion behavior. Explicit radius bounds are deduced by using classical analytic techniques. In addition, AI-assisted numerical experiments are used to verify the sharpness of the theoretical results and to illustrate the dependence of the radius functions on the parameters α and β. Representative numerical values and graphical visualizations are provided.