<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> denote the class of analytic functions <i>f</i> in the unit disc <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {D}=\{z\in \mathbb {C}:\;|z|<1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">D</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mspace width="0.277778em" /> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> normalized by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f(0)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f^{\prime }(0)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In the present article, we consider <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {G}(\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {F}(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> two subclasses of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> which are defined by <Equation ID="Equ41"> <EquationSource Format="TEX">$\begin{aligned} \mathcal {G}(\beta )&=\bigg \{f\in \mathcal {A}:\;\textrm{Re}\;\bigg (1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg )<1+\beta /2\;\;\text{ for }\;\beta >0\bigg \}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="script">G</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">{</mo> </mrow> <mi>f</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> <mo>:</mo> <mspace width="0.277778em" /> <mtext>Re</mtext> <mspace width="0.277778em" /> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">(</mo> </mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">)</mo> </mrow> <mo><</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mspace width="0.333333em" /> <mtext>for</mtext> <mspace width="0.333333em" /> <mspace width="0.277778em" /> <mi>β</mi> <mo>></mo> <mn>0</mn> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">}</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>and <Equation ID="Equ42"> <EquationSource Format="TEX">$\begin{aligned} \mathcal {F}(\alpha )=\bigg \{f\in \mathcal {A}:\;\textrm{Re}\;\bigg (1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg )<\alpha \;\;\text{ for }\;-1/2\le \alpha \le 0\bigg \}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="script">F</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">{</mo> </mrow> <mi>f</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> <mo>:</mo> <mspace width="0.277778em" /> <mtext>Re</mtext> <mspace width="0.277778em" /> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">(</mo> </mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">)</mo> </mrow> <mo><</mo> <mi>α</mi> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mspace width="0.333333em" /> <mtext>for</mtext> <mspace width="0.333333em" /> <mspace width="0.277778em" /> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>≤</mo> <mi>α</mi> <mo>≤</mo> <mn>0</mn> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">}</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>and obtain sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives for functions in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {G}^0(\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">G</mi> </mrow> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {F}^0(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">F</mi> </mrow> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Further, we obtain sharp bounds for distortion and growth theorems for functions in the classes <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {G}^0(\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">G</mi> </mrow> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {F}^0(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">F</mi> </mrow> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Pre-Schwarzian and Schwarzian norm estimates for certain classes of analytic functions

期刊论文
发表日期：

2025年12月5日

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Molla Basir Ahamed,
Vasudevarao Allu,
Rajesh Hossain

摘要

Let $\mathcal {A}$ $A$ denote the class of analytic functions f in the unit disc $\mathbb {D}=\{z\in \mathbb {C}:\;|z|<1\}$ $D = {z \in C : | z | < 1}$ normalized by $f(0)=0$ $f (0) = 0$ and $f^{\prime }(0)=1$ $f^{'} (0) = 1$ . In the present article, we consider $\mathcal {G}(\beta )$ $G (β)$ and $\mathcal {F}(\alpha )$ $F (α)$ two subclasses of $\mathcal {A}$ $A$ which are defined by $\begin{aligned} \mathcal {G}(\beta )&=\bigg \{f\in \mathcal {A}:\;\textrm{Re}\;\bigg (1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg )<1+\beta /2\;\;\text{ for }\;\beta >0\bigg \}, \end{aligned}$ $\begin{matrix} G (β) & = {f \in A : Re (1 + \frac{z f^{″} (z)}{f^{'} (z)}) < 1 + β / 2 for β > 0}, \end{matrix}$ and $\begin{aligned} \mathcal {F}(\alpha )=\bigg \{f\in \mathcal {A}:\;\textrm{Re}\;\bigg (1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg )<\alpha \;\;\text{ for }\;-1/2\le \alpha \le 0\bigg \}, \end{aligned}$ $\begin{matrix} F (α) = {f \in A : Re (1 + \frac{z f^{″} (z)}{f^{'} (z)}) < α for - 1 / 2 \leq α \leq 0}, \end{matrix}$ and obtain sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives for functions in $\mathcal {G}^0(\beta )$ $G^{0} (β)$ and $\mathcal {F}^0(\alpha )$ $F^{0} (α)$ . Further, we obtain sharp bounds for distortion and growth theorems for functions in the classes $\mathcal {G}^0(\beta )$ $G^{0} (β)$ and $\mathcal {F}^0(\alpha )$ $F^{0} (α)$ .