Inequalities of extended (p,q)-beta and confluent hypergeometric function

TL;DR

This paper investigates the mathematical properties of extended (p,q)-beta and confluent hypergeometric functions, establishing their log-convexity, monotonicity, and Turán type inequalities to deepen understanding of their behavior.

Contribution

It introduces new inequalities and properties for extended (p,q)-beta and confluent hypergeometric functions, expanding the theoretical framework of these special functions.

Findings

Proved log-convexity of extended (p,q)-beta functions

Established Turán type inequalities for these functions

Analyzed monotonicity properties of extended (p,q)-confluent hypergeometric functions

Abstract

In this present paper, we establish the log-convexity and Tur\'an type inequalities of extended $(p,q)$-beta functions. Also, we present the log-convexity, the monotonicity and Tur\'an type inequalities for extended $(p,q)$-confluent hypergeometric function by using the inequalities of extended $(p,q)$-beta functions.

Equations

$$\Gamma (z)=\underset{0}{\overset{\infty }{\int }}{t}^{z-1}{e}^{-t}dt,\Re (z)>0.$$

$$\Gamma (z)=\underset{n\to \infty }{\lim }\frac{n!n^{z-1}}{(z{)}_n}$$

$$\displaystyle(\alpha)_{n}=\left\{\begin{array}[]{l}\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+n-1);\quad for\quad n\geq 1,\alpha\neq 0\\ 1\quad if\quad n=0\end{array}\right.$$

$$\Gamma (z+1)=z\Gamma (z)$$

$$(z{)}_n=\frac{\Gamma (z+n)}{\Gamma (z)}.$$

$$B(x,y)=\underset{0}{\overset{1}{\int }}{t}^{x-1}(1-t{)}^{y-1}dt,$$

$$B(x,y)=\frac{\Gamma (x)\Gamma (y)}{\Gamma (x+y)},\Re (x)>0,\Re (y)>0.$$

$${\Gamma }_p(z)=\underset{0}{\overset{\infty }{\int }}{t}^{z-1}{e}^{-t-p{t}^{-1}}dt,$$

$$B(x,y;p)=\underset{0}{\overset{1}{\int }}{t}^{x-1}(1-t{)}^{y-1}{e}^{-\frac{p}{t(1-t)}}dt$$

$$B(x,y;p,q)=B_{p,q}(x,y)=\underset{0}{\overset{1}{\int }}{t}^{x-1}(1-t{)}^{y-1}{e}^{-\frac{p}{t}-\frac{q}{1-t}}dt$$

$$\displaystyle\Phi_{p,q}\Big{(}\beta;\gamma;z\Big{)}=\sum\limits_{n=0}^{\infty}\frac{B(\beta+n;\gamma-\beta;p,q)}{B(\beta,\gamma-\beta)}\frac{z^{n}}{n!}$$

$$\Big{(}p\geq 0,q\geq 0,\Re(\gamma)>\Re(\beta)>0\Big{)}.$$

$$\displaystyle\Phi_{p,q}\Big{(}\beta;\gamma;z\Big{)}=\frac{1}{B(\beta,\gamma-\beta)}\int_{0}^{1}t^{\beta-1}(1-t)^{\gamma-\beta-1}\exp\Big{(}zt-\frac{p}{t}-\frac{q}{1-t}\Big{)}dt$$

$$\big{(}p\geq 0,q\geq 0,\Re(\gamma)>\Re(\beta)>0\Big{)}.$$

$$\underset{a}{\overset{b}{\int }}p(x)f(x)dx\underset{a}{\overset{b}{\int }}p(x)g(x)dx\le \underset{a}{\overset{b}{\int }}p(x)dx\underset{a}{\overset{b}{\int }}p(x)f(x)g(x)dx.$$

$$f(\alpha x_1+(1-\alpha )x_2)\le \alpha f(x_1)+(1-\alpha )f(x_2).$$

$$\log f(\alpha x_1+(1-\alpha )x_2)\le \alpha \log f(x_1)+(1-\alpha )\log f(x_2)=\log (f^{\alpha }(x)f^{1-\alpha }(x_2)).$$

$$f(\alpha x_1+(1-\alpha )x_2)\le f^{\alpha }(x_1)f^{1-\alpha }(x_2).$$

$$|\underset{a}{\overset{b}{\int }}f(x)g(x)dx|\le (\underset{a}{\overset{b}{\int }}|f{|}^{{\theta }_1}dx{)}^{\frac{1}{{\theta }_1}}(\underset{a}{\overset{b}{\int }}|g{|}^{{\theta }_2}dx{)}^{\frac{1}{{\theta }_2}}.$$

$$(x-x_1)(y-y_1)\ge 0,$$

$$B_{p,q}(x,y_1)B_{p,q}(x_1,y)\le B_{p,q}(x_1,y_1)B_{p,q}(x,y),$$

$$f^{\prime }(t)=(x-x_1)t^{x-x_1-1},g^{\prime }(t)=(y_1-y)(1-x{)}^{y-y_1-1}.$$

$$\displaystyle\Big{(}\int_{a}^{b}t^{x-1}(1-t)^{y_{1}-1}\exp\Big{(}-\frac{p}{t}-\frac{q}{1-t}\Big{)}dt\Big{)}\Big{(}\int_{a}^{b}t^{x_{1}-1}(1-t)^{y-1}\exp\Big{(}-\frac{p}{t}-\frac{q}{1-t}\Big{)}dt\Big{)}$$

$$\displaystyle\leq\Big{(}\int_{a}^{b}t^{x_{1}-1}(1-t)^{y_{1}-1}\exp\Big{(}-\frac{p}{t}-\frac{q}{1-t}\Big{)}dt\Big{)}\Big{(}\int_{a}^{b}t^{x-1}(1-t)^{y-1}\exp\Big{(}-\frac{p}{t}-\frac{q}{1-t}\Big{)}dt\Big{)}$$

$$B_{p,q}(x,y_1)B_{p,q}(x_1,y)$$

$$B_{p,q}^2(x,y)-B_{p+a,q+a}(x,y)B_{p-a,q-a}(x,y)\le 0,$$

$$\displaystyle B_{\alpha p_{1}+(1-\alpha)p_{2},\alpha q_{1}+(1-\alpha)q_{2}}(x,y)\leq\Big{(}B_{p,q}(x,y)\Big{)}^{\alpha}\Big{(}B_{p,q}(x,y)\Big{)}^{1-\alpha},$$

$$B_{\alpha p_1+(1-\alpha )p_2,\alpha q_1+(1-\alpha )q_2}(x,y)$$

$$\displaystyle=\int_{0}^{1}t^{x-1}(1-t)^{y-1}\exp\Big{(}\frac{-\alpha p_{1}-(1-\alpha)p_{2}}{t}+\frac{-\alpha q_{1}-(1-\alpha)q_{2}}{1-t}\Big{)}$$

$$\displaystyle=\Big{(}\int_{0}^{1}t^{x-1}(1-t)^{y-1}\exp\Big{(}-\frac{p_{1}}{t}-\frac{q_{1}}{1-t}\Big{)}\Big{)}^{\alpha}$$

$$\displaystyle\times\Big{(}\int_{0}^{1}t^{x-1}(1-t)^{y-1}\exp\Big{(}-\frac{p_{1}}{t}-\frac{q_{1}}{1-t}\Big{)}\Big{)}^{1-\alpha}$$

$$B_{\alpha p_1+(1-\alpha )p_2,\alpha q_1+(1-\alpha )q_2}(x,y)$$

Full text

1 Department of Mathematics, University of Sargodha, Sargodha, Pakistan, email: smjhanda@gmail.com

2 Department of Mathematics, College of Arts & Science-Wadi Addawaser, Prince Sattam bin Abdulaziz University, Saudi Arabia, email: n.sooppy@psau.edu.sa

3 Department of Mathematics, International Islamic University, Islamabad, Pakistan,

emails: gauhar55uom@gmail.com (G. Rahman), marshad_zia@yahoo.com (M. Arshad)

keywords: Extended beta functions, extended hypergeometric functions, log-convexity, Turán-type inequalities.

MSC[2010]: 33B15, 33B99.

1 Introduction

We begin with the classical gamma function

[TABLE]

In another way, it is defined as

[TABLE]

where $(\alpha)_{n}$ is the Pochhammer symbol defined as

[TABLE]

and

[TABLE]

The relation between Pochhammer symbol and gamma function is given below

[TABLE]

The beta function is defined by

[TABLE]

$(\Re(x)>0,\Re(y)>0)$

and

[TABLE]

Chaudhry and Zubair [4] and Chaudhry et al. [5] defined the following extended gamma and beta functions

[TABLE]

$\Re(z)>0,p\geq 0$. When $p=0$, then $\Gamma_{p}$ tends to the classical gamma function $\Gamma$,

and

[TABLE]

(where $\Re(p)>0,\Re(x)>0,\Re(y)>0$) respectively. When $p=0$, then $B(x,y;0)=B(x,y)$. Recently Choi et al. [6] introduced the following extension of extended beta function as

[TABLE]

(where $\Re(p)>0,\Re(q)>0,\Re(x)>0,\Re(y)>0$).

It is clear that when $p=q$, then (1.6) reduces to the well known extended beta function (1.5). Similarly if $p=q=0$, then (1.6) reduces to the classical beta function (1.2). In the same paper, they also defined the following extension of extended confluent hypergeometric function by

[TABLE]

[TABLE]

The integral representations of extension of extended confluent hypergeometric function is given by

[TABLE]

[TABLE]

Note that for $p=q$, the series (1.7) respectively reduces to the extended confluent hypergeometric series. Similarly for $p=q=0$ the series (1.7) respectively reduces to the classical confluent hypergeometric series.

2 Main results: Inequalities of extended $(p,q)$-beta function

In this section, we establish some inequalities which involve extended $(p,q)$-beta functions by using some natural inequalities [10]. For this continuation of our study, we recall the following well-known Chebychev’s integral inequality and Hölder-Rogers inequality.

Lemma 2.1**.**

(see [7, 8]) Let the functions $f,g:[a,b]\subseteq{R}\rightarrow{R}$ are asynchronous for all $x\in[a,b]$ and $p(x):[a,b]\subseteq{R}\rightarrow{R}$ is a positive integrable function, then

[TABLE]

Definition 2.1**.**

In [3], a function $f:(a,b)\rightarrow{R}$ is said to be convex if for any $x_{1},x_{2}\in(a,b)$ and $\alpha\in(0,1)$

[TABLE]

It shows that when we move from $x_{1}$ to $x_{2}$, the line joining the points $(x_{1},f(x_{1}))$ and $(x_{2},f(x_{2}))$ lies always above the graph of $f$.

Definition 2.2**.**

A function $f$ is said to be a log-convex if $f>0$ and $\log f$ is convex i.e., for all $x_{1},x_{2}\in I$ (where I is an interval) and $\alpha\in(0,1)$, we have

[TABLE]

This implies that

[TABLE]

Lemma 2.2**.**

(Hölder inequality [12]) If $\theta_{1}$ and $\theta_{2}$ are positive real numbers such that $\frac{1}{\theta_{1}}+\frac{1}{\theta_{2}}=1$, then the following inequality holds for integrable functions $f,g:[a,b]\rightarrow{R}$:

[TABLE]

Theorem 1**.**

If $x,y,x_{1},y_{1}$ are positive real numbers satisfying the condition

[TABLE]

then for the extended $(p,q)$-beta function, we have the inequality

[TABLE]

Proof.

Consider the mappings $f,g,h:[0,1]\rightarrow[0,\infty)$ given by

$f(t)=t^{x-x_{1}}$, $g(t)=(1-t)^{y-y_{1}}$ and h(t)=t^{x_{1}-1}(1-t)^{y_{1}-1}\exp\Big{(}-\frac{p}{t}-\frac{q}{1-t}\Big{)}.

Now, differentiation of $f$ and $g$ gives

[TABLE]

This show that $f$ and $g$ have the same monotonicity on $[0,1]$.

Applying the Chebyshev’s integral inequality (2.1), for the above defined functions $f$, $g$ and $h$, we have

[TABLE]

which implies that,

[TABLE]

which completes the desired proof. ∎

Theorem 2**.**

The function $(p,q)\mapsto B_{p,q}(x,y)$ is log convex on $(0,\infty)$ for each $x,y>0$. Moreover, the function $B_{p,q}(x,y)$ satisfy the following Turán type inequality

[TABLE]

for all real $a$.

Proof.

From the definition of log-convexity, it will be sufficient to prove that

[TABLE]

for $\alpha\in[0,1]$, $p_{1},p_{2},q_{1},q_{2}>0$ and for a fixed $x,y>0$. Obviously, (2.8) is true for $\alpha=0$ and $\alpha=1$. Assume that $\alpha\in(0,1)$, then it follows from (1.6) that

[TABLE]

Let $\theta_{1}=\frac{1}{\alpha}$ and $\theta_{2}=\frac{1}{(1-\alpha)}$. Clearly $\theta_{1}>1$ and $\theta_{1}+\theta_{2}=\theta_{1}\theta_{2}$. Thus applying the Hölder-Rogers inequality (2.4) for integrals in (2.9) gives

[TABLE]

This implies that $(p,q)\mapsto B_{p,q}(x,y)$ is log convex on $(0,\infty)$.

Now, taking $\alpha=\frac{1}{2}$, $p_{1}=p-a$, $p_{2}=p+a$, and $q_{1}=q-a$, $q_{2}=q+a$, the inequality (2.10) yields

[TABLE]

∎

Theorem 3**.**

The function $(x,y)\mapsto B_{p,q}(x,y)$ is logarithmic convex on $(0,\infty)\times(0,\infty)$, for all $p,q\geq 0$. In particular

[TABLE]

Proof.

Let $(x_{1},y_{1}),(x_{2},y_{2})\in(0,\infty)^{2}$, and $c,d\geq 0$ with $c+d=1$, then we have

[TABLE]

Applying the definition of $(p,q)$-extended beta function on the right hand side of inequality (2.11), we have

[TABLE]

Again by considering $\theta_{1}=\frac{1}{c}$, $\theta_{2}=\frac{1}{d}$, we can use the Hölder-Rogers inequality for above integrals and it follows

[TABLE]

This shows the logarithmic convexity of extended $(p,q)$-beta function $B_{p,q}(x,y)$ on $(0,\infty)^{2}$.

For $c=d=\frac{1}{2}$, the above inequality reduces to

[TABLE]

Let $x,y>0$ be such that $\min_{a\in{R}}(x+a,x-a)>0$, then by taking $x_{1}=x+a$, $x_{2}=x+a$, $y_{1}=y+b$ and $y_{2}=y-b$ in (2.12), we get

[TABLE]

for all $p,q\geq 0$. ∎

3 Inequalities for $(p,q)$-extended confluent hypergeometric function

In this section, we present the log-convexity and Turán type inequality for extended confluent hypergeometric function defined in (1.7). For this continuation, we recall the following well-known lemma.

Lemma 3.1**.**

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

Note that the above lemma is valid only if both $f$ and $g$ are both even or both odd functions.

Theorem 4**.**

*Let $\beta\geq 0$ and $\gamma,\delta>0$, then the following assertions for extended $(p,q)$-confluent hypergeometric function are true.

(i) For $\gamma\geq\delta$, the function x\mapsto\Phi_{p,q}\Big{(}\beta;\gamma;x\Big{)}/\Phi_{p,q}\Big{(}\beta;\delta;x\Big{)} is increasing on $(0,\infty)$.

(ii) For $\gamma\geq\delta$, we have

\delta\Phi_{p,q}\Big{(}\beta+1;\gamma+1;x\Big{)}\Phi_{p,q}\Big{(}\beta;\delta;x\Big{)}\geq\gamma\Phi_{p,q}(\beta;\gamma;x)\Phi_{p,q}\Big{(}\beta+1;\delta+1;x\Big{)}.

(iii) The function x\mapsto\Phi_{p,q}\Big{(}\beta;\gamma;x\Big{)} is log-convex on ${R}$.

(iv) The function (p,q)\mapsto\Phi_{p,q}\Big{(}\beta;\gamma;x\Big{)} is log convex on $(0,\infty)$ for fixed $x>0$.

(v) Let $\sigma>0$. then the function*

[TABLE]

is decreasing on $(0,\infty)$ for fixed $\gamma,x>0$.

Proof.

From the definition of (1.7), it follows that

[TABLE]

If we denote $f_{n}=a_{n}(c)/a_{n}(d)$, then

[TABLE]

Now take $x=\beta+n$, $y=\delta-\beta$, $x_{1}=\beta+n+1$, $y_{1}=\gamma-\beta$ in (2.6). Since $(x-x_{1})(y-y_{1})=\gamma-\delta\geq 0$, it follows from Theorem 1 that

[TABLE]

this is equivalent to say that $\{f_{n}\}$ is an increasing sequence and hence with the aid of Lemma 3.1, we observe that x\mapsto\Phi_{p,q}\Big{(}\beta;\gamma;x\Big{)}/\Phi_{p,q}\Big{(}\beta;\delta;x\Big{)} is increasing on $(0,\infty)$.

To prove the assertion (ii), we recall the following well-known identity from [6]:

[TABLE]

Since the increasing properties of x\mapsto\Phi_{p,q}\Big{(}\beta;\gamma;x\Big{)}/\Phi_{p,q}\Big{(}\beta,\delta;x\Big{)} is equivalent to the following inequality

[TABLE]

This together with (3.2) implies

[TABLE]

This implies that

[TABLE]

which prove the assertion. The log-convexity of x\mapsto\Phi_{p,q}\Big{(}\beta;\gamma;x\Big{)} can be prove by using the integral representation of extended $(p,q)$-confluent hypergeometric function as given in (1.8) and by applying the Hölder-Rogers inequality for integrals as follows:

[TABLE]

This prove that x\mapsto\Phi_{p,q}\Big{(}\beta;\gamma;x\Big{)} is log-convex for a fixed $x>0$. For the case when $x<0$, then the assertion immediately follows from the identity (see [6]):

[TABLE]

Since, the infinite sum of log-convex functions is log-convex for $x>0$. Thus, the log-convexity of (p,q)\mapsto\Phi_{p,q}\Big{(}\beta;\gamma;x\Big{)} is equivalent to prove that $(p,q)\mapsto B(\beta+n,\gamma-\beta)$ is log-convex on $(0,\infty)$ and for non-negative integer $n$. From Theorem 2, it is clear that $(p,q)\mapsto B(\beta+n,\gamma-\beta)$ is log-convex for $\gamma>\beta>0$ and hence assertion (iv) is true.

Now, let $\beta^{\prime}\geq\beta$ and set h(t)=t^{\beta^{\prime}-1}(1-t)^{\gamma-\beta^{\prime}-1}\exp\Big{(}xt-\frac{p}{t}-\frac{q}{1-t}\Big{)}, f(t)=\Big{(}\frac{t}{1-t}\Big{)}^{\beta-\beta^{\prime}} and g(t)=\Big{(}\frac{t}{1-t}\Big{)}^{\sigma}. Then using the integral representation (1.8) of extended confluent hypergeometric function, we have

[TABLE]

One can easily determine that for $\beta^{\prime}\geq\beta$, the function $f$ is decreasing when $\sigma\geq 0$ and the function $g$ is increasing . Since $h$ is non negative function for $t\in[0,1]$. Thus, by reverse Chebyshev’s reverse inequality (2.1), it follows that

[TABLE]

This together with (3) implies

[TABLE]

which is equivalent to say that the function

[TABLE]

is decreasing on $(0,\infty)$. ∎

Remark 3.1**.**

In particular, the following decreasing property of extended $(p,q)$-confluent hypergeometric function

[TABLE]

is equivalent to the following inequality

[TABLE]

When $p=q$, then the above inequality will reduce to the inequality recently proved by [11]. Similarly, when $p=q=0$, then the above inequality reduces to the inequality of confluent hypergeometric which is an improved version of Theorem 4(b) given in [9].

4 conclusion

In this paper, we introduced inequalities for extended $(p,q)$-beta and $(p,q)$-confluent hypergeometric function defined by Choi et al. [6]. Throughout in this paper, if we take $p=q$ then we get the inequalities of extended beta function and extended confluent hypergeometric function recently introduced by Mondal [11]. Similarly if we take $p=q=0$, then the newly defined inequalities for extended $(p,q)$-beta function will reduce to the inequalities of classical beta function (see [1, 7]).