{"title":"Geometric characterization of the generalized Lommel–Wright function in the open unit disc","authors":"Hanaa M. Zayed, Teodor Bulboacă","doi":"10.1186/s13660-024-03108-2","DOIUrl":null,"url":null,"abstract":"The present investigation aims to examine the geometric properties of the normalized form of the combination of generalized Lommel–Wright function $\\mathfrak{J}_{\\lambda ,\\mu}^{\\nu ,m}(z):=\\Gamma ^{m}(\\lambda +1) \\Gamma (\\lambda +\\mu +1)2^{2\\lambda +\\mu}z^{1-(\\nu /2)-\\lambda} \\mathcal{J}_{\\lambda ,\\mu }^{\\nu ,m}(\\sqrt{z})$ , where the function $\\mathcal{J}_{\\lambda ,\\mu}^{\\nu ,m}$ satisfies the differential equation $\\mathcal{J}_{\\lambda ,\\mu}^{\\nu ,m}(z):=(1-2\\lambda -\\nu )J_{ \\lambda ,\\mu}^{\\nu ,m}(z)+z (J_{\\lambda ,\\mu }^{\\nu ,m}(z) )^{\\prime}$ with $$ J_{\\nu ,\\lambda}^{\\mu ,m}(z)= \\biggl(\\frac{z}{2} \\biggr)^{2\\lambda + \\nu} \\sum_{k=0}^{\\infty} \\frac{(-1)^{k}}{\\Gamma ^{m} (k+\\lambda +1 )\\Gamma (k\\mu +\\nu +\\lambda +1 )} \\biggl(\\frac{z}{ 2} \\biggr)^{2k} $$ for $\\lambda \\in \\mathbb{C}\\setminus \\mathbb{Z}^{-}$ , $\\mathbb{Z}^{-}:= \\{ -1,-2,-3,\\ldots \\}$ , $m\\in \\mathbb{N}$ , $\\nu \\in \\mathbb{C}$ , and $\\mu \\in \\mathbb{N}_{0}:=\\mathbb{N}\\cup \\{0\\}$ . In particular, we employ a new procedure using mathematical induction, as well as an estimate for the upper and lower bounds for the gamma function inspired by Li and Chen (J. Inequal. Pure Appl. Math. 8(1):28, 2007), to evaluate the starlikeness and convexity of order α, $0\\leq \\alpha <1$ . Ultimately, we discuss the starlikeness and convexity of order zero for $\\mathfrak{J}_{\\lambda ,\\mu} ^{\\nu ,m}$ , and it turns out that they are useful to extend the range of validity for the parameter λ to $\\lambda \\geq 0$ where the main concept of the proofs comes from some technical manipulations given by Mocanu (Libertas Math. 13:27–40, 1993). Our results improve, complement, and generalize some well-known (nonsharp) estimates.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"4 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Inequalities and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1186/s13660-024-03108-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The present investigation aims to examine the geometric properties of the normalized form of the combination of generalized Lommel–Wright function $\mathfrak{J}_{\lambda ,\mu}^{\nu ,m}(z):=\Gamma ^{m}(\lambda +1) \Gamma (\lambda +\mu +1)2^{2\lambda +\mu}z^{1-(\nu /2)-\lambda} \mathcal{J}_{\lambda ,\mu }^{\nu ,m}(\sqrt{z})$ , where the function $\mathcal{J}_{\lambda ,\mu}^{\nu ,m}$ satisfies the differential equation $\mathcal{J}_{\lambda ,\mu}^{\nu ,m}(z):=(1-2\lambda -\nu )J_{ \lambda ,\mu}^{\nu ,m}(z)+z (J_{\lambda ,\mu }^{\nu ,m}(z) )^{\prime}$ with $$ J_{\nu ,\lambda}^{\mu ,m}(z)= \biggl(\frac{z}{2} \biggr)^{2\lambda + \nu} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{\Gamma ^{m} (k+\lambda +1 )\Gamma (k\mu +\nu +\lambda +1 )} \biggl(\frac{z}{ 2} \biggr)^{2k} $$ for $\lambda \in \mathbb{C}\setminus \mathbb{Z}^{-}$ , $\mathbb{Z}^{-}:= \{ -1,-2,-3,\ldots \}$ , $m\in \mathbb{N}$ , $\nu \in \mathbb{C}$ , and $\mu \in \mathbb{N}_{0}:=\mathbb{N}\cup \{0\}$ . In particular, we employ a new procedure using mathematical induction, as well as an estimate for the upper and lower bounds for the gamma function inspired by Li and Chen (J. Inequal. Pure Appl. Math. 8(1):28, 2007), to evaluate the starlikeness and convexity of order α, $0\leq \alpha <1$ . Ultimately, we discuss the starlikeness and convexity of order zero for $\mathfrak{J}_{\lambda ,\mu} ^{\nu ,m}$ , and it turns out that they are useful to extend the range of validity for the parameter λ to $\lambda \geq 0$ where the main concept of the proofs comes from some technical manipulations given by Mocanu (Libertas Math. 13:27–40, 1993). Our results improve, complement, and generalize some well-known (nonsharp) estimates.
期刊介绍:
The aim of this journal is to provide a multi-disciplinary forum of discussion in mathematics and its applications in which the essentiality of inequalities is highlighted. This Journal accepts high quality articles containing original research results and survey articles of exceptional merit. Subject matters should be strongly related to inequalities, such as, but not restricted to, the following: inequalities in analysis, inequalities in approximation theory, inequalities in combinatorics, inequalities in economics, inequalities in geometry, inequalities in mechanics, inequalities in optimization, inequalities in stochastic analysis and applications.