Inequalities for the modified k-Bessel function

TL;DR

This paper investigates inequalities and monotonicity properties of the generalized k-Bessel functions, including their ratios and log-convexity, providing new insights into their mathematical behavior.

Contribution

It introduces new monotonicity and convexity results for the generalized k-Bessel functions and their ratios, expanding understanding of their inequalities.

Findings

Monotonicity of the ratio of different order k-Bessel functions

Log-convexity of the k-Bessel functions with respect to order

Monotonicity of the ratio between k-Bessel and k-confluent hypergeometric functions

Abstract

The article considers the generalized k-Bessel functions and represents it as Wright functions. Then we study the monotonicity properties of the ratio of two different orders k- Bessel functions, and the ratio of the k-Bessel and the m-Bessel functions. The log-convexity with respect to the order of the k-Bessel also given. An investigation regarding the monotonicity of the ratio of the k-Bessel and k-confluent hypergeometric functions are discussed.

Equations

$${\Gamma }_k(x):=\underset{n\to \infty }{\lim }\frac{n!k^n(n^k{)}^{\frac{x}{k}-1}}{(x{)}_{n,k}},k>0,$$

$${\Gamma }_k(x)={\int }_0^{\infty }{t}^{x-1}{e}^{-\frac{t^k}{k}}dt,\mathrm{Re}(x)>0.$$

$$J_{k,\nu }^{\gamma ,\lambda }\left(x\right):=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{n,k}}{{\Gamma }_k\left(\lambda n+\upsilon +1\right)}\frac{{\left(-1\right)}^n{\left(x/2\right)}^n}{{\left(n!\right)}^2},$$

$$I_{k,\nu }^{\gamma ,\lambda }\left(x\right):=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{n,k}}{{\Gamma }_k\left(\lambda n+\upsilon +1\right)}\frac{{\left(x/2\right)}^n}{{\left(n!\right)}^2},$$

$$\underset{k,\lambda ,\gamma \to 1}{\lim }{I}_{k,\nu }^{\gamma ,\lambda }\left(x\right)=\sum _{n=0}^{\infty }\frac{1}{\Gamma \left(n+\upsilon +1\right)}\frac{{\left(x/2\right)}^n}{n!}={\left(\frac{2}{x}\right)}^{\frac{\nu }{2}}{I}_{\nu }(\sqrt{2x}),$$

$${\mathtt{W}}_{k,\nu ,c}^{\gamma ,\lambda }(x):=\sum _{n=0}^{\infty }\frac{(\gamma {)}_{n,k}}{{\Gamma }_k(\lambda n+\nu +1)}\frac{(-c{)}^n(x/2{)}^n}{{\left(n!\right)}^2},c\in \mathbb{R}.$$

$${}_p{F}_q(a_1,\dots ,a_p;c_1,\dots ,c_q;z)=\sum _{k=0}^{\infty }\frac{(a_1{)}_k\cdots (a_p{)}_k}{(c_1{)}_k\cdots (c_q{)}_k(1{)}_k}{z}^k,|z|<1,$$

$${}_{p}\psi_{q}(z)={}_{p}\psi_{q}\left[\begin{array}[]{c}(a_{i},\alpha_{i})_{1,p}\\ (b_{j},\beta_{j})_{1,q}\end{array}\bigg{|}z\right]=\displaystyle\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{p}\Gamma(a_{i}+\alpha_{i}k)}{\prod_{j=1}^{q}\Gamma(b_{j}+\beta_{j}k)}\frac{z^{k}}{k!},$$

$$\sum _{j=1}^q{\beta }_j-\sum _{i=1}^p{\alpha }_i>-1$$

$$\mathtt{W}_{k,\nu,c}^{\gamma,\lambda}(x)=\frac{1}{k^{\frac{\nu+k+1}{k}}\Gamma\left(\frac{\gamma}{k}\right)}{}_{1}\psi_{2}\left[\begin{array}[]{ccc}\left(\frac{\gamma}{k},1\right)&\\ \left(\frac{\nu+1}{k},\frac{\gamma}{k}\right)&(1,1)\end{array}\bigg{|}-\frac{cx}{2k^{\frac{\lambda}{k}-1}}\right]$$

$${\mathtt{W}}_{k,\nu ,c}^{\gamma ,\lambda }(x)$$

$$=\frac{1}{k^{\frac{\nu +k+1}{k}}\Gamma \left(\frac{\gamma }{k}\right)}\sum _{n=0}^{\infty }\frac{\Gamma \left(\frac{\gamma }{k}+n\right)}{\Gamma \left(\frac{\lambda }{k}n+\frac{\nu +1}{k}\right)\Gamma \left(\frac{\gamma }{k}\right)}\frac{(-c{)}^n}{\Gamma (n+1)\Gamma (n+1)}{\left(\frac{x}{2k^{\frac{\lambda }{k}-1}}\right)}^n$$

$$\displaystyle=\frac{1}{k^{\frac{\nu+k+1}{k}}\Gamma\left(\frac{\gamma}{k}\right)}{}_{1}\psi_{2}\left[\begin{array}[]{ccc}\left(\frac{\gamma}{k},1\right)&\\ \left(\frac{\nu+1}{k},\frac{\gamma}{k}\right)&(1,1)\end{array}\bigg{|}-\frac{cx}{2k^{\frac{\lambda}{k}-1}}\right]$$

$${\mathtt{I}}_{k,\nu }^{\gamma ,\lambda }(x)=\sum _{n=0}^{\infty }{a}_n(\nu )x^n\text{and}{\mathtt{I}}_{k,\nu }^{\gamma ,\lambda }(x)=\sum _{n=0}^{\infty }{a}_n(\mu )x^n,$$

$$a_n(\nu )=\frac{(\gamma {)}_{n,k}}{{\Gamma }_k(\lambda n+\nu +1)(n!{)}^2{2}^n}\text{and}{a}_n(\mu )=\frac{(\gamma {)}_{n,k}}{{\Gamma }_k(\lambda n+\mu +1)(n!{)}^2{2}^n}$$

$$f(t):=\frac{{\Gamma }_k(\lambda t+\mu +1)}{{\Gamma }_k(\lambda t+\nu +1)}.$$

$$\frac{f^{\prime }(t)}{f(t)}=\lambda ({\Psi }_k(\lambda t+\mu +1)-{\Psi }_k(\lambda t+\nu +1)).$$

$${\Psi }_k(t)=\frac{\log (k)-{\gamma }_1}{k}-\frac{1}{t}+\sum _{n=1}^{\infty }\frac{t}{nk(nk+t)}$$

$${\Psi }_k^{\prime }(t)=\sum _{n=0}^{\infty }\frac{1}{(nk+t{)}^2},k>0\text{and}t>0.$$

$$a_n^k\left(\nu \right)=\frac{{\left(\gamma \right)}_{n,k}}{{\Gamma }_k\left(\lambda n+\nu +1\right){\left(n!\right)}^2}\text{and}{a}_n^m\left(\nu \right)=\frac{{\left(\gamma \right)}_{n,m}}{{\Gamma }_m\left(\lambda n+\nu +1\right){\left(n!\right)}^2},$$

$$d_n$$

$$=\frac{{\Gamma }_k\left(\gamma +nk\right){\Gamma }_m\left(\lambda n+\nu +1\right)}{{\Gamma }_k\left(\gamma +nm\right){\Gamma }_k\left(\lambda n+\nu +1\right)}.$$

$$f(y):=\frac{{\Gamma }_k\left(\gamma +yk\right){\Gamma }_m\left(\lambda y+\nu +1\right)}{{\Gamma }_k\left(\gamma +ym\right){\Gamma }_k\left(\lambda y+\nu +1\right)}$$

$$\frac{f^{\prime }(y)}{f(y)}=k{\Psi }_k(\gamma +yk)+\lambda {\Psi }_m\left(\lambda y+\nu +1\right)-m{\Psi }_m(\gamma +ym)-\lambda {\Psi }_k\left(\lambda y+\nu +1\right)$$

$$\displaystyle\frac{f^{\prime}(y)}{f(y)}\geq\lambda\big{(}\Psi_{k}(\nu+1+yk)-\Psi_{k}\left(\lambda y+\nu+1\right)\big{)}+m\big{(}\Psi_{m}\left(\lambda y+\nu+1\right)-\Psi_{m}(\nu+1+ym)\big{)}\geq 0.$$

$$\nu \mapsto a_n^k\left(\nu \right):=\frac{{\left(\gamma \right)}_{n,k}{\Gamma }_k\left(\nu +1\right)}{{\Gamma }_k\left(\lambda n+\nu +1\right){\left(n!\right)}^2}$$

$$\frac{\partial }{\partial \nu }\log \left(a_n^k\left(\nu \right)\right)={\Psi }_k\left(\nu +1\right)-{\Psi }_k\left(\lambda n+\nu +1\right).$$

$$\frac{{\partial }^2}{\partial {\nu }^2}\log \left(a_n^k\left(\nu \right)\right)$$

$$=\sum _{r=0}^{\infty }\frac{1}{(rk+\nu +1{)}^2}-\sum _{r=0}^{\infty }\frac{1}{(rk+\lambda n+\nu +1{)}^2}$$

$$=\sum _{r=0}^{\infty }\frac{\lambda n(2rk+\lambda n+2\nu +2)}{(rk+\nu +1{)}^2(rk+\lambda n+\nu +1{)}^2}>0,$$

$$a_n(\nu )=\frac{(\gamma {)}_{n,k}}{{\Gamma }_k(\lambda n+\nu +1)(n!{)}^2{2}^n}\text{and}{d}_{n,k}\left(a,c\right)=\frac{{\left(a\right)}_{n,k}}{{\left(c\right)}_{n,k}n!}$$

$$w_n=\frac{a_n\left(\nu \right)}{d_{n,k}\left(a,c\right)}$$

$${\rho }_k\left(x\right)=\frac{{\Gamma }_k\left(\gamma +xk\right){\Gamma }_k\left(c+xk\right)}{{\Gamma }_k\left(\lambda x+\nu +1\right){\Gamma }_k\left(a+xk\right)2^x\Gamma (x+1)}.$$

Full text

1. Introduction

One of the generalization of the classical gamma function $\Gamma$ studied in [4] is defined by the limit formula

[TABLE]

where $(x)_{n,k}:=x(x+k)(x+2k)\ldots(x+(n-1)k)$ is called $k$-Pochhammer symbol. The above $k-$gamma function also have an integral representation as

[TABLE]

Properties of the $k$-gamma functions have been studies by many researchers [8, 9, 10, 11, 6]. Follwoing properties are required in sequel:

• (i)

$\Gamma_{k}\left(x+k\right)=x\Gamma_{k}\left(x\right)$

• (ii)

$\Gamma_{k}\left(x\right)=k^{\frac{x}{k}-1}\Gamma\left(\frac{x}{k}\right)$

• (iii)

$\Gamma_{k}\left(k\right)=1$

• (iv)

$\Gamma_{k}\left(x+nk\right)=\Gamma_{k}(x)(x)_{n,k}$

Motivated with the above generalization of the $k$-gamma functions, Romero et. al.[1] introduced the $k-$Bessel function of the first kind defined by the series

[TABLE]

where $k\in\mathbb{R^{+}}$; $\alpha,\lambda,\gamma,\upsilon\in C$; $\operatorname{Re}(\lambda)>0$ and $\operatorname{Re}(\upsilon)>0$. They also established two recurrence relations for $J_{k,\nu}^{\gamma,\lambda}$.

In this article, we are considering the following function:

[TABLE]

Since

[TABLE]

the classical modified Bessel functions of first kind. In this sense, we can call $I_{k,\nu}^{\gamma,\lambda}$ as the modified $k$-Bessel functions of first kind. In fact, we can express both $J_{k,\nu}^{\gamma,\lambda}$ and $I_{k,\nu}^{\gamma,\lambda}$ together in

[TABLE]

We can termed $\mathtt{W}_{k,\nu}^{\gamma,\lambda}$ as the generalized $k$-Bessel function.

First we study the representation formulas for $\mathtt{W}_{k,\nu}^{\gamma,\lambda}$ in term of the classical Wright functions. Then we will study about the monotonicity and log-convexity properties of $I_{k,\nu}^{\gamma,\lambda}$.

2. Representation formula for the generalized $k$-Bessel function

The generalized hypergeometric function ${}_{p}F_{q}(a_{1},\ldots,a_{p};c_{1},\ldots,c_{q};x)$, is given by the power series

[TABLE]

where the $c_{i}$ can not be zero or a negative integer. Here $p$ or $q$ or both are allowed to be zero. The series $(\ref{eqn:gen-hyp-func})$ is absolutely convergent for all finite $z$ if $p\leq q$ and for $|z|<1$ if $p=q+1$. When $p>q+1$, then the series diverge for $z\not=0$ and the series does not terminate.

The generalized Wright hypergeometric function ${}_{p}\psi_{q}(z)$ is given by the series

[TABLE]

where $a_{i},b_{j}\in\mathbb{C}$, and real $\alpha_{i},\beta_{j}\in\mathbb{R}$ ($i=1,2,\ldots,p;j=1,2,\ldots,q$). The asymptotic behavior of this function for large values of argument of $z\in\mathbb{C}$ were studied in [13, 14] and under the condition

[TABLE]

in literature [18, 19]. The more properties of the Wright function are investigated in [14, 15, 16].

Now we will give the representation of the generalized $k$-Bessel functions in terms of the Wright and generalized hypergeometric functions.

Proposition 2.1**.**

Let, $k\in\mathbb{R}$ and $\lambda,\gamma,\nu\in\mathbb{C}$ such that $\operatorname{Re}(\lambda)>0,\operatorname{Re}(\nu)>0.$ Then

[TABLE]

Proof.

Using the relations $\Gamma_{k}\left(x\right)=k^{\frac{x}{k}-1}\Gamma\left(\frac{x}{k}\right)$ and $\Gamma_{k}\left(x+nk\right)=\Gamma_{k}(x)(x)_{n,k}$, the generalized $k$-Bessel functions defined in $\eqref{eqn-genb}$ can be rewrite as

[TABLE]

Hence the result follows. ∎

3. Monotonicty and log-convexity properties

This section discuss the monotonicity and log-convexity properties for the modified $k$-Bessel functions $\mathtt{W}_{k,\nu,-1}^{\gamma,\lambda}(x)=\mathtt{I}_{k,\nu}^{\gamma,\lambda}(x)$.

Following lemma due to Biernacki and Krzyż [7] will be required.

Lemma 3.1**.**

[7]** Consider the power series $f(x)=\sum_{k=0}^{\infty}a_{k}x^{k}$ and $g(x)=\sum_{k=0}^{\infty}b_{k}x^{k}$, where $a_{k}\in\mathbb{R}$ and $b_{k}>0$ for all $k$. Further suppose that both series converge on $|x|<r$. If the sequence $\{a_{k}/b_{k}\}_{k\geq 0}$ is increasing (or decreasing), then the function $x\mapsto f(x)/g(x)$ is also increasing (or decreasing) on $(0,r)$.

The above lemma still holds when both $f$ and $g$ are even, or both are odd functions.

Theorem 3.1**.**

The following results holds true for the modified $k$-Bessel functions.

1. (1)

For $\mu\geq\nu>-1$, the function $x\mapsto\mathtt{I}_{k,\mu}^{\gamma,\lambda}(x)/\mathtt{I}_{k,\nu}^{\gamma,\lambda}(x)$ is increasing on $(0,\infty)$ for some fixed $k>0$. 2. (2)

If $k\geq\lambda\geq m>0$, the function $x\mapsto\mathtt{I}_{k,\nu}^{\gamma,\lambda}(x)/\mathtt{I}_{m,\nu}^{\gamma,\lambda}(x)$ is increasing on $(0,\infty)$ for some fixed $\nu>-1$ and $\gamma\geq\nu+1$. 3. (3)

The function $\nu\mapsto\mathcal{I}_{k,\nu}^{\gamma,\lambda}(x)$ is log-convex on $(0,\infty)$ for some fixed $k,\gamma>0$ and $x>0$. Here, $\mathcal{I}_{k,\nu}^{\gamma,\lambda}(x):=\Gamma_{k}(\nu+1)\mathtt{I}_{k,\nu}^{\gamma,\lambda}(x)$. 4. (4)

Suppose that $\lambda\geq k>0$ and $\nu>-1$. Then

1. (a)

The function $x\mapsto\mathtt{I}_{k,\nu}^{\gamma,\lambda}(x)/\Phi_{k}\left(a,c;x\right)$ is decreasing on $(0,\infty)$ for $a\geq c>0$ and $0<\gamma\leq\nu+1$. Here, $\Phi_{k}\left(a;c;x\right)$ is the $k$-confluent hypergeometric functions. 2. (b)

The function $x\mapsto\mathtt{I}_{k,\nu}^{\gamma,\lambda}(x)/\Phi_{k}\left(\gamma;\lambda;x/2\right)$ is decreasing on $(0,1)$ for $\gamma>0$ and $0<k\leq\lambda\leq\nu+1$. 3. (c)

The function $x\mapsto\mathtt{I}_{k,\nu}^{\gamma,\lambda}(x)/\Phi_{k}\left(\gamma;\lambda;x/2\right)$ is decreasing on $[1,\infty)$ for $\gamma>0$ and $0<k\leq\min\{\lambda,\nu+1\}$.

Proof.

${\bf(1)}$ Form (1.4) it follows that

[TABLE]

where

[TABLE]

Consider the function

[TABLE]

Then the logarithmic differentiation yields

[TABLE]

Here, $\Psi_{k}=\Gamma_{k}^{\prime}/\Gamma_{k}$ is the $k$-digamma functions studied in [5] and defined by

[TABLE]

where $\gamma_{1}$ is the Euler-Mascheroni’s constant.

A calculation yields

[TABLE]

Clearly, $\Psi_{k}$ is increasing on $(0,\infty)$ and hence $f^{\prime}(t)>0$ for all $t\geq 0$ if $\mu\geq\nu>-1$. This, in particular, implies that the sequence $\{d_{n}\}_{n\geq 0}=\{a_{n}(\nu)/a_{n}(\mu)\}_{n\geq 0}$ is increasing and hence the conclusion follows from Lemma 3.1.

(2). This result also follows from Lemma 3.1 if the sequence $\{d_{n}\}_{n\geq 0}=\{a_{n}^{k}(\nu)/a_{n}^{m}(\mu)\}_{n\geq 0}$ is increasing for $k\geq m>0$. Here,

[TABLE]

which together with the identity $\Gamma_{k}\left(x+nk\right)=\Gamma_{k}(x)(x)_{n,k}$ gives

[TABLE]

Now to show that $\{d_{n}\}$ is increase, consider the function

[TABLE]

The logarithmic differentiation of $f$ yields

[TABLE]

If $\gamma\geq\nu+1$ and $k\geq\lambda\geq m$, then (3.3) can be rewrite as

[TABLE]

This conclude that $f$, and consequently the sequence $\{d_{n}\}_{n\geq 0}$, is increasing. Finally the result follows from the Lemma 3.1.

(3). It is known that sum of the log-convex functions is log-convex. Thus, to prove the result it is enough to show that

[TABLE]

is log-convex.

A logarithmic differentiation of $a_{n}(\nu)$ with respect to $\nu$ yields

[TABLE]

This along with (3.2) gives

[TABLE]

for all $n\geq 0$, $k>0$ and $\nu>-1$. Thus, $\nu\mapsto a_{n}^{k}\left(\nu\right)$ is log-convex and hence the conclusion.

(4). Denote $\Phi_{k}\left(a,c;x\right)=\sum_{n=0}^{\infty}c_{n,k}(a,c)x^{n}$ and $\mathtt{I}_{k,\nu}^{\gamma,\lambda}(x)=\sum_{n=0}^{\infty}a_{n}(\nu)x^{n},$ where

[TABLE]

with $v>-1$ and $a,c,\lambda,\gamma,k>0.$ To apply Lemma 3.1, consider the sequence $\left\{w_{n}\right\}_{n\geq 0}$ defined by

[TABLE]

where

[TABLE]

In view of the increasing properties of $\Psi_{k}$ on $(0,\infty)$, and

[TABLE]

it follows that for $a\geq c>0$, $\lambda\geq k$ and $\nu+1\geq\gamma$, the function $\rho$ is decreasing on $\left(0,\infty\right)$ and thus the sequence $\left\{w_{n}\right\}_{n\geq 0}$ also decreasing. Finally the conclusion for $(a)$ follows from the Lemma 3.1.

In the case $(b)$ and $(c)$, the sequence $\{w_{n}\}$ reduces to

[TABLE]

where

[TABLE]

Now as in the proof of part (a)

[TABLE]

if $\nu+1+\lambda x\geq\lambda+xk$. Now for $x\in(0,1)$, this inequality holds if $0<k\leq\lambda\leq\nu+1$, while for $x\geq 1$, it is required that $k\leq\min\{\lambda,\nu+1\}.$ ∎