The Euler–Mascheroni sequence <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\big \{\gamma _n-\gamma \big \}_{n\ge 1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">{</mo> </mrow> <msub> <mi>γ</mi> <mi>n</mi> </msub> <mo>-</mo> <mi>γ</mi> <msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> is defined by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma _n = -\log n + \sum _{l=1}^n 1/l$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mi>n</mi> </msub> <mo>=</mo> <mo>-</mo> <mo>log</mo> <mi>n</mi> <mo>+</mo> <msubsup> <mo>∑</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <mn>1</mn> <mo stretchy="false">/</mo> <mi>l</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\in {\mathbb N }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma =\lim _{n\rightarrow \infty }\gamma _n= 0.57721\,56649\dots $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </msub> <msub> <mi>γ</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.57721</mn> <mspace width="0.166667em" /> <mn>56649</mn> <mo>⋯</mo> </mrow> </math></EquationSource> </InlineEquation> is the Euler–Mascheroni constant. This paper deals with the analytic function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f(z)=\sum _{n=1}^\infty \big ( \frac{1}{2n}-\gamma _n + \gamma \big ) z^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <mo>-</mo> <msub> <mi>γ</mi> <mi>n</mi> </msub> <mo>+</mo> <mi>γ</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and it is shown that f(z) is universally starlike. In addition it is shown that the functions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f(z)=\sum _{n=1}^\infty \big ( \frac{1}{2n}-\gamma _n+\gamma \big ) z^n/n^\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <mo>-</mo> <msub> <mi>γ</mi> <mi>n</mi> </msub> <mo>+</mo> <mi>γ</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <msup> <mi>z</mi> <mi>n</mi> </msup> <mo stretchy="false">/</mo> <msup> <mi>n</mi> <mi>α</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> are also universally starlike for every <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

On geometric properties of the generating function for the Euler–Mascheroni sequence

期刊论文
发表日期：

2025年12月17日

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Raquel Pezoa,
Luis Salinas,
Claudio E. Torres

摘要

The Euler–Mascheroni sequence $\big \{\gamma _n-\gamma \big \}_{n\ge 1}$ ${γ_{n} - γ}_{n \geq 1}$ is defined by $\gamma _n = -\log n + \sum _{l=1}^n 1/l$ $γ_{n} = - log n + \sum_{l = 1}^{n} 1 / l$ , $n\in {\mathbb N }$ $n \in N$ , and $\gamma =\lim _{n\rightarrow \infty }\gamma _n= 0.57721\,56649\dots $ $γ = {lim}_{n \to \infty} γ_{n} = 0.57721 56649 \dots$ is the Euler–Mascheroni constant. This paper deals with the analytic function $f(z)=\sum _{n=1}^\infty \big ( \frac{1}{2n}-\gamma _n + \gamma \big ) z^n$ $f (z) = \sum_{n = 1}^{\infty} (\frac{1}{2 n} - γ_{n} + γ) z^{n}$ and it is shown that f(z) is universally starlike. In addition it is shown that the functions $f(z)=\sum _{n=1}^\infty \big ( \frac{1}{2n}-\gamma _n+\gamma \big ) z^n/n^\alpha $ $f (z) = \sum_{n = 1}^{\infty} (\frac{1}{2 n} - γ_{n} + γ) z^{n} / n^{α}$ are also universally starlike for every $\alpha \ge 0$ $α \geq 0$ .