Some lower bounds for the quotients of normalized error function and their partial sums

Basem Aref Frasin

Archivum Mathematicum (2025)

  • Issue: 2, page 73-83
  • ISSN: 0044-8753

Abstract

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The purpose of the present paper is to determine lower bounds for ℜ ℰ k f ( z ) ( ℰ k f ) m ( z ) , ℜ ( ℰ k f ) m ( z ) ℰ k f ( z ) , ℜ ℰ k ' f ( z ) ( ℰ k f ) m ' ( z ) and ℜ ( ℰ k f ) m ' ( z ) ℰ k ' f ( z ) , where ℰ k f is the generalized normalized error function of the form ℰ k f z = z + ∑ n = 2 ∞ - 1 n - 1 ( n - 1 k + 1 ) n - 1 ! z n and ( ℰ k f ) m its partial sum. Furthermore, we give lower bounds for ℜ 𝕀 ℰ k f ( z ) ( 𝕀 ℰ k f ) m ( z ) and ℜ ( 𝕀 ℰ k f ) m ( z ) 𝕀 ℰ k f ( z ) , where 𝕀 ℰ k f is the Alexander transform of ℰ k f . Several examples of the main results are also considered.

How to cite

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Frasin, Basem Aref. "Some lower bounds for the quotients of normalized error function and their partial sums." Archivum Mathematicum (2025): 73-83. <http://eudml.org/doc/299993>.

@article{Frasin2025,
abstract = {The purpose of the present paper is to determine lower bounds for $\mathfrak \{R\}\left\rbrace \frac\{\mathcal \{E\}_\{k\}f(z)\}\{(\mathcal \{E\}_\{k\}f)_\{m\}(z)\}\right\lbrace $, $\mathfrak \{R\}\left\rbrace \frac\{(\mathcal \{E\}_\{k\}f)_\{m\}(z)\}\{\mathcal \{E\}_\{k\}f(z)\}\right\lbrace , \mathfrak \{R\}\left\rbrace \frac\{\mathcal \{E\}_\{k\}^\{\prime \}f(z)\}\{(\mathcal \{E\}_\{k\}f)_\{m\}^\{\prime \}(z)\}\right\lbrace $ and $\mathfrak \{R\}\left\rbrace \frac\{(\mathcal \{E\}_\{k\}f)_\{m\}^\{\prime \}(z)\}\{\mathcal \{E\}_\{k\}^\{\prime \}f(z)\}\right\lbrace $, where $\mathcal \{E\}_\{k\}f$ is the generalized normalized error function of the form $\mathcal \{E\}_\{k\}f\left( z\right) =z+\sum _\{n=2\}^\{\infty \}\frac\{\left( -1\right) ^\{n-1\}\}\{(\left( n-1\right) k+1)\left( n-1\right) !\}z^\{n\}$ and $(\mathcal \{E\}_\{k\}f)_\{m\}$ its partial sum. Furthermore, we give lower bounds for $\mathfrak \{R\}\left\rbrace \frac\{\mathbb \{I\}\left[ \mathcal \{E\}_\{k\}f\right] (z)\}\{(\mathbb \{I\}\left[ \mathcal \{E\}_\{k\}f\right] )_\{m\}(z)\}\right\lbrace $ and $\mathfrak \{R\}\left\rbrace \frac\{(\mathbb \{I\}\left[ \mathcal \{E\}_\{k\}f\right] )_\{m\}(z)\}\{\mathbb \{I\}\left[ \mathcal \{E\}_\{k\}f\right] (z)\}\right\lbrace $, where $\mathbb \{I\}\left[ \mathcal \{E\}_\{k\}f\right] $ is the Alexander transform of $\mathcal \{E\}_\{k\}f$. Several examples of the main results are also considered.},
author = {Frasin, Basem Aref},
journal = {Archivum Mathematicum},
keywords = {partial sums; analytic functions; generalized error function},
language = {eng},
number = {2},
pages = {73-83},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Some lower bounds for the quotients of normalized error function and their partial sums},
url = {http://eudml.org/doc/299993},
year = {2025},
}

TY - JOUR
AU - Frasin, Basem Aref
TI - Some lower bounds for the quotients of normalized error function and their partial sums
JO - Archivum Mathematicum
PY - 2025
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
IS - 2
SP - 73
EP - 83
AB - The purpose of the present paper is to determine lower bounds for $\mathfrak {R}\left\rbrace \frac{\mathcal {E}_{k}f(z)}{(\mathcal {E}_{k}f)_{m}(z)}\right\lbrace $, $\mathfrak {R}\left\rbrace \frac{(\mathcal {E}_{k}f)_{m}(z)}{\mathcal {E}_{k}f(z)}\right\lbrace , \mathfrak {R}\left\rbrace \frac{\mathcal {E}_{k}^{\prime }f(z)}{(\mathcal {E}_{k}f)_{m}^{\prime }(z)}\right\lbrace $ and $\mathfrak {R}\left\rbrace \frac{(\mathcal {E}_{k}f)_{m}^{\prime }(z)}{\mathcal {E}_{k}^{\prime }f(z)}\right\lbrace $, where $\mathcal {E}_{k}f$ is the generalized normalized error function of the form $\mathcal {E}_{k}f\left( z\right) =z+\sum _{n=2}^{\infty }\frac{\left( -1\right) ^{n-1}}{(\left( n-1\right) k+1)\left( n-1\right) !}z^{n}$ and $(\mathcal {E}_{k}f)_{m}$ its partial sum. Furthermore, we give lower bounds for $\mathfrak {R}\left\rbrace \frac{\mathbb {I}\left[ \mathcal {E}_{k}f\right] (z)}{(\mathbb {I}\left[ \mathcal {E}_{k}f\right] )_{m}(z)}\right\lbrace $ and $\mathfrak {R}\left\rbrace \frac{(\mathbb {I}\left[ \mathcal {E}_{k}f\right] )_{m}(z)}{\mathbb {I}\left[ \mathcal {E}_{k}f\right] (z)}\right\lbrace $, where $\mathbb {I}\left[ \mathcal {E}_{k}f\right] $ is the Alexander transform of $\mathcal {E}_{k}f$. Several examples of the main results are also considered.
LA - eng
KW - partial sums; analytic functions; generalized error function
UR - http://eudml.org/doc/299993
ER -

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