<p>The primary objective of this paper is to establish the sharp estimates of the pre-Schwarzian norm for functions <i>f</i> in the classes <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {S}^(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {C}(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varphi (z)=1/(1-z)^s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$0<s\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varphi (z)=(1+sz)^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>s</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$0<s\le 1/\sqrt{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {S}^(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {C}(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are the Ma–Minda type starlike and Ma–Minda type convex classes associated with <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>, respectively.</p>

Pre-Schwarzian norm estimate for certain classes of analytic functions

期刊论文
发表日期：

2026年2月27日

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Vasudevarao Allu,
Raju Biswas,
Rajib Mandal

摘要

The primary objective of this paper is to establish the sharp estimates of the pre-Schwarzian norm for functions f in the classes $\mathcal {S}^*(\varphi )$ $S^{*} (φ)$ and $\mathcal {C}(\varphi )$ $C (φ)$ when $\varphi (z)=1/(1-z)^s$ $φ (z) = 1 / {(1 - z)}^{s}$ with $0<s\le 1$ $0 < s \leq 1$ and $\varphi (z)=(1+sz)^2$ $φ (z) = {(1 + s z)}^{2}$ with $0<s\le 1/\sqrt{2}$ $0 < s \leq 1 / \sqrt{2}$ , where $\mathcal {S}^*(\varphi )$ $S^{*} (φ)$ and $\mathcal {C}(\varphi )$ $C (φ)$ are the Ma–Minda type starlike and Ma–Minda type convex classes associated with $\varphi $ $φ$ , respectively.