Abstract
In the paper, the authors present some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function via some classical inequalities such as Chebychev’s inequality for synchronous (or asynchronous, respectively) mappings, give a new proof of the log-convexity of the extended gamma function by using the Hölder inequality, and introduce a Turán type mean inequality for the Kummer confluent k-hypergeometric function.
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1 Introduction
The gamma function Γ can be defined [24, 28, 31, 32] by
Alternatively, it can also be defined [20] by
where \((z)_{n}\) for \(z\ne0\) is the Pochhammer symbol defined [27] as
The relation between \((z)_{n}\) and \(\Gamma(z)\) is
The beta function \(B(x,y)\) can be defined [18, 21, 22] by
and can be expressed by
In 1995, Chaudhry and Zubair [4] introduced the extended gamma function,
If \(b=0\), then \(\Gamma_{b}\) becomes the classical gamma function Γ.
In 1997, Chaudhry et al. [3] introduced the extended beta function,
It is clear that \(B_{0}(x,y)=B(x,y)\).
In 2009, Barnard et al. [1] established three inequalities
and
where \(A(\alpha,\beta)=\frac{\alpha+\beta}{2}\) and \(G(\alpha,\beta )=\sqrt{\alpha\beta}\) are the arithmetic and geometric means and
is the Kummer confluent hypergeometric function [25, 28].
The Kummer confluent hypergeometric k-function is defined by
where
for \(n\ge1\) and \(k>0\) with \((a)_{0,k}=1\) is the Pochhammer k-symbol, which can also be rewritten as
and the gamma k-function \(\Gamma_{k}(a)\) is defined [6] by
In 2012, Mubeen [15] introduced the k-analogue of Kummer’s transformation as
In Sect. 2, we prepare two lemmas. In Sect. 3, we discuss applications of some integral inequalities such as Chebychev’s integral inequality. In Sect. 4, we prove the logarithmic convexity of the extended gamma function. In the last section, we introduce a mean inequality of Turán type for the Kummer confluent hypergeometric k-function.
2 Lemmas
In order to obtain our main results, we need the following lemmas.
Lemma 2.1
(Chebychev’s integral inequality [7, 8, 12, 23])
Let \(f,g,h:I\subseteq\mathbb{R}\to\mathbb{R}\) be mappings such that \(h(x)\ge0\), \(h(x)f(x)g(x)\), \(h(x)f(x)\), and \(h(x)g(x)\) are integrable on I. If \(f(x)\) and \(g(x)\) are synchronous (or asynchronous, respectively) on I, that is,
for all \(x,y\in I\), then
Lemma 2.2
(Hölder’s inequality [29, 30])
Let p and q be positive real numbers such that \(\frac{1}{p}+\frac {1}{q}=1\) and \(f,g:[c,d]\to\mathbb{R}\) be integrable functions. Then
3 Inequalities involving the extended gamma function via Chebychev’s integral inequality
In this section, we prove some inequalities involving the extended gamma function via Chebychev’s integral inequality in Lemma 2.1.
Theorem 3.1
Let m, p and r be positive real numbers such that \(p>r>-m\). If \(r(p-m-r)\gtreqless0\), then
Proof
Let us define the mappings \(f,g,h:[0,\infty)\rightarrow[0,\infty)\) given by
If \(r(p-m-r)\gtreqless0\), then we can claim that the mappings f and g are synchronous (asynchronous) on \((0,\infty)\). Thus, by applying Chebychev’s inequality on \(I=(0,\infty)\) to the functions f, g and h defined above, we can write
This implies that
By (1.1), we acquire the required inequality (3.1). □
Corollary 3.1
If \(p>0\) and \(q\in\mathbb{R}\) with \(\vert q \vert < p\), then
Proof
By setting \(m=p\) and \(r=q\) in Theorem 3.1, we obtain \(r(p-m-r)=-q^{2}\leq0\) and then the inequality (3.1) provides the desired Corollary 3.1. □
Theorem 3.2
If \(m, n>0\) are similarly (oppositely) unitary, then
Proof
Consider the mappings \(f,g,h:[0,\infty)\rightarrow[0,\infty)\) defined by
Now if the condition \((m-1)(n-1)\gtreqless0\) holds, then Chebychev’s integral inequality applied to the functions f, g, and h means
This implies that
By the definition of the extended gamma function, we have
or
The required proof is complete. □
Corollary 3.2
If \(b=0\), then
Theorem 3.3
If m and n are positive real numbers such that m and n are similarly (oppositely) unitary, then
Proof
Consider the mappings \(f,g,h:[0,\infty)\rightarrow[0,\infty)\) defined by
If the conditions of Theorem 3.1 hold, then the mappings f and g are synchronous (asynchronous) on \([0,\infty)\). Thus, by applying Chebychev’s integral inequality in Lemma 2.1 to the functions f, g and h defined above, we have
This implies that
Thus by the definition of extended gamma function, we have
The required proof is complete. □
Corollary 3.3
If \(b=0\), then
4 Log-convexity of the extended gamma function
It is well known that, if \(f>0\) and lnf is convex, then f is said to be a logarithmically convex function. Every logarithmically convex must be convex. See [16] and [19, Remark 1.9]. In this section, we verify the log-convexity of extended gamma function.
Theorem 4.1
The extended gamma function \(\Gamma_{b}:(0,\infty)\to \mathbb{R}\) is logarithmically convex.
Proof
Let p and q be positive numbers such that \(\frac{1}{p}+\frac {1}{q}=1\). Since
see [5], letting \(\lambda=\frac{1}{p}\) and \((1-\lambda)=\frac {1}{q}\) leads to
As a result, the function \(\Gamma_{b}\) is logarithmically convex. □
5 A mean inequality of the Turán type for the Kummer confluent hypergeometric k-function
In this section, we present a mean inequality involving the confluent hypergeometric k-function. For this purpose, we consider the relation
Theorem 5.1
For \(a,b, k>0\) and \(v\in\mathbb{N}\) with \(a,b\ge v-k\), the inequality
is valid for all nonzero \(x\in\mathbb{R}\).
First proof
Assume that \(x>0\). For \(c\ne0,-1,-2,\ldots\) , define
and
From (5.1), it follows that
where
Accordingly, by the Cauchy product, we have
where
If s is even, then
where \(\lceil x\rceil\) denotes the ceiling function whose value is the greatest integer not more than x. Similarly, if s is odd,
Accordingly,
Carefully simplifying gives
where \(h_{k}(x)=\frac{(x)_{s-r,k}}{(x)_{s,k}}\). For \(x>0\) and \(s-r>r\), that is, \([\frac{s-1}{2}]\ge r\), the logarithmic derivatives of \(h_{k}\) is
where \(\psi_{k}=\frac{\Gamma_{k}'}{\Gamma_{k}}\) is the digamma k-function (see [6, 11, 16]). Hence, the function \(h_{k}\) is increasing under the condition stated. This fact together with the aid of (5.3) and (5.4) yields
where \(a\ge v\ge0\), \(x>0\), \(c+k>0\), and \(c\ne0\). Consequently, from (5.5), it follows that
is positive for \(a\ge v\ge v-k\ge v-2k\ge\cdots\ge0\) and \(f_{0,k}(x)=0\). Now replacing v by \(v-k\) shows that
Therefore, the function \(f_{v,k}\) is absolutely monotonic on \((0,\infty )\), that is, \(f_{v,k}^{(\ell)}(x)>0\) for \(\ell=0,1,2,\ldots\) . This proves Theorem 5.1 for the case \(x>0\).
Now suppose that \(x<0\), \(a,b>0\), and \(v\in\mathbb{N}\) with \(a,b\ge v-k\). Since \(\phi_{k}(a,c,x)\) is symmetric in a and b, by interchanging a and b in Theorem 5.1, we obtain
By using Kummer’s transformation (1.2), we have
Thus, Theorem 5.1 also holds for \(x<0\). □
Second proof
Since
it follows that
Replacing a and b by \(\frac{a}{k}\) and \(\frac{b}{k}\), respectively, gives Theorem 5.1. □
Corollary 5.1
If \(a>0\) and \(c+k>0\) with \(c\ne0\), then the inequality
holds for any \(v\in\mathbb{N}\) with \(a\ge v-k\).
Proof
This follows directly from the proof of Theorem 5.1 and the fact that Eq. (5.6) holds under the conditions \(c+k>0\) and \(c\ne0\). □
Corollary 5.2
If \(v\in\mathbb{N}\) and \(a,b\ge v\), then
for all nonzero \(x\in\mathbb{R}\), where A and G are, respectively, the arithmetic and geometric means.
Proof
First assume \(x\ge0\) and \(a,b\ge v\) for \(v\in\mathbb{N}\). Then the left hand side inequality in (5.7) is a direct consequence of the facts that
for \(s=0,1\) and
for \(s\ge2\). Hence, by induction, we have
For \(x\ge0\), the right hand side inequality in (5.7) follows from taking square root of (5.2). The proof of Corollary 5.2 for \(x\ge0\) is thus complete.
Now assume \(x<0\) with \(a,b\ge v\). Interchanging a and b in (5.7) one arrives at
Making use of the k-analogue of Kummer’s transformation and the homogeneity of A and G acquires
Consequently, Theorem (5.7) also follows for \(x<0\). □
Remark 5.1
In Sect. 5, we have established a Turán type and mean inequality for k-analogue of the Kummer confluent hypergeometric function. If we let \(k\to1\), then we can conclude to the corresponding inequalities of the confluent hypergeometric function.
Remark 5.2
In [2], some inequalities of the Turán type for confluent hypergeometric functions of the second kind were also discovered.
Remark 5.3
By the way, we note that Refs. [9, 10, 13, 14, 26, 32, 33] belong to the same series in which inequalities and complete monotonicity for functions involving the gamma function \(\Gamma(x)\) and the logarithmic function \(\ln(1+x)\) were discussed.
Remark 5.4
This paper is a slightly revised version of the preprint [17].
6 Conclusions
In this paper, we present some inequalities involving the extended gamma function \(\Gamma_{b}(z)\) via some classical inequalities such as Chebychev’s inequality for synchronous (or asynchronous, respectively) mappings, give a new proof of the log-convexity of the extended gamma function \(\Gamma_{b}(z)\) by using the Hölder inequality, and introduce a Turán type mean inequality for the Kummer confluent k-hypergeometric function \(\phi(z)\).
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Nisar, K.S., Qi, F., Rahman, G. et al. Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function. J Inequal Appl 2018, 135 (2018). https://doi.org/10.1186/s13660-018-1717-8
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DOI: https://doi.org/10.1186/s13660-018-1717-8