<p>Let <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\gamma \in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\beta \in (-\frac{\pi }{2}, \frac{\pi }{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> be a holomorphic function such that <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\Phi (0)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\Re (\Phi (\zeta ))>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℜ</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> on the open unit disk in <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>. A class of <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-spirllike mappings <i>f</i> of real order <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> on the unit ball <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\mathbb {B_{X}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">B</mi> <mi mathvariant="double-struck">X</mi> </msub> </math></EquationSource> </InlineEquation> in a complex Banach space <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\mathbb {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">X</mi> </math></EquationSource> </InlineEquation> is defined, which has a <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-parametric representation and satisfies that <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$x=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is zero of order <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$k+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$f(x)-x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>. Kinds of versions of sharp distortion results are established over this class. Our works extend the distortion theorem of holomorphic functions in <InlineEquation ID="IEq26"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation> to the case in <InlineEquation ID="IEq27"> <EquationSource Format="TEX">$\mathbb {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">X</mi> </math></EquationSource> </InlineEquation>.</p>

Distortion theorems of $\beta $-spirllike mappings of real order $\gamma $ which have a $\Phi $-parametric representation in $\mathbb {C}^{n}$ $^{*}$

期刊论文
发表日期：

2025年7月1日

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Fang Yu,
Luyi Zeng,
Liangpeng Xiong

摘要

Let $\gamma \in (0,1]$ $γ \in (0, 1]$ , $\beta \in (-\frac{\pi }{2}, \frac{\pi }{2})$ $β \in (- \frac{π}{2}, \frac{π}{2})$ and $\Phi $ $Φ$ be a holomorphic function such that $\Phi (0)=1$ $Φ (0) = 1$ and $\Re (\Phi (\zeta ))>0$ $ℜ (Φ (ζ)) > 0$ on the open unit disk in $\mathbb {C}$ $C$ . A class of $\beta $ $β$ -spirllike mappings f of real order $\gamma $ $γ$ on the unit ball $\mathbb {B_{X}}$ $B_{X}$ in a complex Banach space $\mathbb {X}$ $X$ is defined, which has a $\Phi $ $Φ$ -parametric representation and satisfies that $x=0$ $x = 0$ is zero of order $k+1$ $k + 1$ of $f(x)-x$ $f (x) - x$ . Kinds of versions of sharp distortion results are established over this class. Our works extend the distortion theorem of holomorphic functions in $\mathbb {C}$ $C$ to the case in $\mathbb {X}$ $X$ .

Distortion theorems of \(\beta \)-spirllike mappings of real order \(\gamma \) which have a \(\Phi \)-parametric representation in \(\mathbb {C}^{n}\) \(^{*}\)

摘要