For the differential equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(1-z)^nw''+a(1-z)^mw' +bw=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> <msup> <mi>w</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>b</mi> <mi>w</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> (with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n>m\ge 0,\, a\in \mathbb {R},\, b\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mi>m</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em" /> <mi>a</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>b</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>), the existence of analytical solutions in the unit disk of the form <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f(z)=F(1/(1-z))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where F is an entire transcendental function, is studied. For such a function F, the growth, starlikeness, convexity, and close-to-convexity are investigated.

ON ANALYTIC SOLUTIONS OF THE DIFFERENTIAL EQUATION OF SHAH TYPE IN THE UNIT DISK. GROWTH AND GEOMETRIC PROPERTIES

期刊论文
发表日期：

2025年11月1日

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Myroslav M. Sheremeta,
Yurii S. Trukhan

摘要

For the differential equation $(1-z)^nw''+a(1-z)^mw' +bw=0$ ${(1 - z)}^{n} w^{''} + a {(1 - z)}^{m} w^{'} + b w = 0$ (with $n>m\ge 0,\, a\in \mathbb {R},\, b\in \mathbb {R}$ $n > m \geq 0, a \in R, b \in R$ ), the existence of analytical solutions in the unit disk of the form $f(z)=F(1/(1-z))$ $f (z) = F (1 / (1 - z))$ , where F is an entire transcendental function, is studied. For such a function F, the growth, starlikeness, convexity, and close-to-convexity are investigated.