Convexity properties related to Gauss hypergeometric function
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916160099844096 |
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| author | Bouali, Mohamed |
| author_facet | Bouali, Mohamed |
| contents | We investigate the convexity property on $(0,1)$ of the functions $φ_{a,b,c}$ and $1/φ_{a,b,c}$, where $$φ_{a,b,c}(x)= \frac{c-\log(1-x)}{\,_2F_1(a,b,a+b,x)},$$ whenever $a,b\geq 0$ and $a+b\leq 1$. We Show that $φ_{a,b,c}$ (respectively $1/φ_{a,b,c}$) is strictly convex on $(0,1)$ if and only if $c\leq -2γ-ψ(a)-ψ(b),$ (respectively $c\geqα_0$) and $φ_{a,b,c}$ (respectively $1/φ_{a,b,c}$) is strictly concave on $(0,1)$ if and only if $c\geq c(a,b)$ (respectively $c\in[δ_-,δ_+]$), where $ψ$ is the Polygamma function. This generalizes some problems posed by Yang and Tian and complete the study of convexity properties of functions studied by the author in [bouali]. As applications of the convexity and concavity, we establish among other inequalities, that for all $x\in(0,1)$, $a,b\in[0,1]$, $a+b\leq 1$ and $c\geq c(a,b)$ $$c+\frac{Γ(a)Γ(b)}{Γ(a+b)}\leq \frac{c-\log(1-x)}{\,_2F_1(a,b,a+b,x)}+\frac{c-\log(x)}{\,_2F_1(a,b,a+b,1-x)}\leq\frac{(2c+2\log 2)}{\,_2{F}_1(a,b;a+b;1/2)},$$ and for all $x\in(0,1)$, $a,b\in[0,1]$, $a+b\leq 1$ and $c\in [δ_-,δ_+]$ $$\frac1c+\frac{Γ(a+b)}{Γ(a)Γ(b)}\leq \frac{\,_2F_1(a,b,a+b,x)}{c-\log(1-x)}+\frac{\,_2F_1(a,b,a+b,1-x)}{c-\log(x)}\leq\frac{\,_2{F}_1(a,b;a+b;1/2)}{(2c+2\log 2)}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09695 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convexity properties related to Gauss hypergeometric function Bouali, Mohamed General Mathematics We investigate the convexity property on $(0,1)$ of the functions $φ_{a,b,c}$ and $1/φ_{a,b,c}$, where $$φ_{a,b,c}(x)= \frac{c-\log(1-x)}{\,_2F_1(a,b,a+b,x)},$$ whenever $a,b\geq 0$ and $a+b\leq 1$. We Show that $φ_{a,b,c}$ (respectively $1/φ_{a,b,c}$) is strictly convex on $(0,1)$ if and only if $c\leq -2γ-ψ(a)-ψ(b),$ (respectively $c\geqα_0$) and $φ_{a,b,c}$ (respectively $1/φ_{a,b,c}$) is strictly concave on $(0,1)$ if and only if $c\geq c(a,b)$ (respectively $c\in[δ_-,δ_+]$), where $ψ$ is the Polygamma function. This generalizes some problems posed by Yang and Tian and complete the study of convexity properties of functions studied by the author in [bouali]. As applications of the convexity and concavity, we establish among other inequalities, that for all $x\in(0,1)$, $a,b\in[0,1]$, $a+b\leq 1$ and $c\geq c(a,b)$ $$c+\frac{Γ(a)Γ(b)}{Γ(a+b)}\leq \frac{c-\log(1-x)}{\,_2F_1(a,b,a+b,x)}+\frac{c-\log(x)}{\,_2F_1(a,b,a+b,1-x)}\leq\frac{(2c+2\log 2)}{\,_2{F}_1(a,b;a+b;1/2)},$$ and for all $x\in(0,1)$, $a,b\in[0,1]$, $a+b\leq 1$ and $c\in [δ_-,δ_+]$ $$\frac1c+\frac{Γ(a+b)}{Γ(a)Γ(b)}\leq \frac{\,_2F_1(a,b,a+b,x)}{c-\log(1-x)}+\frac{\,_2F_1(a,b,a+b,1-x)}{c-\log(x)}\leq\frac{\,_2{F}_1(a,b;a+b;1/2)}{(2c+2\log 2)}.$$ |
| title | Convexity properties related to Gauss hypergeometric function |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2403.09695 |