Abstract
In the article, we deal with the monotonicity of the function \(x\rightarrow[ (x^{p}+a )^{1/p}-x]/I_{p}(x)\) on the interval \((0, \infty)\) for \(p>1\) and \(a>0\), and present the necessary and sufficient condition such that the double inequality \([ (x^{p}+a )^{1/p}-x]/a< I_{p}(x)<[ (x^{p}+b )^{1/p}-x]/b\) for all \(x>0\) and \(p>1\), where \(I_{p}(x)=e^{x^{p}}\int_{x}^{\infty}e^{-t^{p}}\,dt\) is the incomplete gamma function.
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1 Introduction
Let \(a>0\) and \(x>0\). Then the classical gamma function \(\Gamma(x)\), incomplete gamma function \(\Gamma(a, x)\) and psi function \(\psi(x)\) are defined by
respectively. It is well known that the identities
hold for all \(x, p>0\).
Recently the bounds for the integral \(\int_{x}^{\infty}e^{-t^{p}}\,dt\) or \(\int_{0}^{x}e^{-t^{p}}\,dt\) have attracted the attention of many researchers. In particular, many remarkable inequalities for bounding both integrals can be found in the literature [1–12]. Let
Then \(I_{2}(x)\) is actually the Mills ratio and it has been investigated by many researchers [13–19], and the functions \(I_{3}(x)\) and \(I_{4}(x)\) can be used to research the heat transfer problem [20] and electrical discharge in gases [21], respectively.
Komatu [22] and Pollak [23] proved that the double inequality
holds for all \(x>0\).
In [24], Gautschi proved that the double inequality
holds for all \(x>0\) and \(p>1\), where
An application of inequality (1.3) was given in [25]. Alzer [26] proved that the double inequality
holds for all \(x>0\) and \(p>0\) with \(p\neq1\) if and only if \(\alpha\geq\max\{1, \Gamma^{-p}(1+1/p)\}\) and \(\beta\leq\min\{1, \Gamma^{-p}(1+1/p)\}\).
Motivated by inequality (1.3), in the article we deal with the monotonicity of the function
and prove that the double inequality
holds for all \(x>0\) and \(p>1\) if and only if \(a\geq2\) and \(b\leq a_{0}=\Gamma^{p/(1-p)}(1+1/p)\).
2 Lemmas
In order to prove our main results, we need to introduce an auxiliary function at first.
Let \(-\infty\leq a< b\leq\infty\), f and g be differentiable on \((a,b)\), and \(g'\neq0\) on \((a,b)\). Then the function \(H_{f, g}\) [27, 28] is defined by
Lemma 2.1
(See [28], Theorem 9)
Let \(\infty\leq a< b\leq\infty\), f and g be differentiable on \((a,b)\) with \(f(b^{-})=g(b^{-})=0\) and \(g^{\prime}(x)<0\) on \((a,b)\), \(H_{f, g}\) be defined by (2.1), and there exists \(\lambda\in(a, b)\) such that \(f^{\prime}(x)/g^{\prime}(x)\) is strictly increasing on \((a, \lambda)\) and strictly decreasing on \((\lambda, b)\). Then the following statements are true:
-
(1)
if \(H_{f, g}(a^{+})\geq0\), then \(f(x)/g(x)\) is strictly decreasing on \((a, b)\);
-
(2)
if \(H_{f, g}(a^{+})<0\), then there exists \(x_{0}\in(a, b)\) such that \(f(x)/g(x)\) is strictly increasing on \((a, x_{0})\) and strictly decreasing on \((x_{0}, b)\).
Lemma 2.2
(See [29], Theorem 1.25)
Let \(-\infty< a< b<\infty\), \(f,g:[a,b]\rightarrow{\mathbb{R}}\) be continuous on \([a,b]\) and differentiable on \((a,b)\), and \(g'(x)\neq 0\) on \((a,b)\). If \(f^{\prime}(x)/g^{\prime}(x)\) is increasing (decreasing) on \((a,b)\), then so are the functions
If \(f^{\prime}(x)/g^{\prime}(x)\) is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.3
The inequality
holds for all \(x\in(0, 1)\).
Proof
We clearly see that inequality (2.2) is equivalent to
for \(x\in(0, 1)\).
Let
Then simple computations lead to
for \(x\in(0, 1)\), where \(\gamma=0.5772\ldots\) is the Euler-Mascheroni constant.
Therefore, inequality (2.3) follows easily from (2.4)-(2.6). □
Lemma 2.4
The function \(\Gamma^{1/x}(1+x)\) is strictly increasing on \((0, \infty)\), and the double inequality
holds for all \(x\in(0, 1)\).
Proof
Let
Then simple computations lead to
for \(x\in(0, \infty)\), and
for \(x\in(0, 1)\).
It follows from (2.8), (2.10), (2.12), and Lemma 2.2 that \(\varphi(x)\) and \(e^{\varphi(x)}=\Gamma^{1/x}(1+x)\) is strictly increasing on \((0, \infty)\).
Inequality (2.13) leads to the conclusion that the function \(\phi(x)\) is strictly concave on the interval \((0, 1)\) and the inequality
holds for all \(x\in(0, 1)\).
Therefore, \(\phi(x)>0\) and the first inequality of (2.7) holds for all \(x\in(0, 1)\) follows from (2.9), (2.11), and (2.14). While the second inequality of (2.7) can be derived from the monotonicity of the function \(\Gamma^{1/x}(1+x)\) on the interval \((0, 1)\). □
Lemma 2.5
Let \(p>1\) and \(x>0\). Then the function \(a\rightarrow [ (x^{p}+a )^{1/p}-x ]/a\) is strictly decreasing on \((0, \infty)\).
Proof
Let
Then we clearly see that
for all \(p>1\), \(x>0\) and \(a>0\).
Therefore, Lemma 2.5 follows easily from Lemma 2.2 and (2.15)-(2.17). □
Lemma 2.6
Let \(p>1\), \(a>0\) and \(x>0\), \(H_{f, g}(x)\) be defined by (2.1), and \(f_{1}(x)\) and \(g_{1}(x)\) be defined by
respectively. Then \(H_{f_{1}, g_{1}}(0^{+})=\Gamma(1+1/p)-a^{1/p}\).
Proof
Let
Then from (2.18) and (2.19) one has
It follows from (2.1), (2.20), and (2.21) that
□
3 Main results
Theorem 3.1
Let \(p>1\), \(a>0\), \(x>0\) and \(R(x)\) be defined by (1.5). Then the following statements are true:
-
(1)
if \(a\geq2\), then \(R(x)\) is strictly increasing on \((0, \infty)\);
-
(2)
if \(a\leq\Gamma^{p}(1+1/p)\), then \(R(x)\) is strictly decreasing on \((0, \infty)\);
-
(3)
if \(\Gamma^{p}(1+1/p)< a<2\), then there exists \(x_{0}\in(0, \infty)\) such that \(R(x)\) is strictly increasing on \((0, x_{0})\) and strictly decreasing on \((x_{0}, \infty)\).
Proof
Let \(f_{1}(x)\), \(g_{1}(x)\), \(u=u(x)\in(1, \infty)\) be defined by (2.18) and (2.19), and \(h(u)\) and \(h_{1}(u)\) be defined by
Then from (1.2), (1.5), (2.18), (2.21), (3.1), (3.2), and Lemma 2.4 we have
for \(p>1\).
We divide the proof into four cases.
Case 1: \(a\geq2\). Then from (3.4)-(3.7) we clearly see that the function \(f^{\prime}_{1}(x)/g^{\prime}_{1}(x)\) is strictly increasing on \((0, \infty)\). Therefore, \(R(x)\) is strictly increasing on \((0, \infty)\) follows from Lemma 2.2 and (3.3) together with the monotonicity of the function \(f^{\prime}_{1}(x)/g^{\prime}_{1}(x)\) on the interval \((0, \infty)\) and \(f_{1}(\infty)=g_{1}(\infty)=0\).
Case 2: \(a\leq1/p\). Then from (3.4)-(3.8) we clearly see that the function \(f^{\prime}_{1}(x)/g^{\prime}_{1}(x)\) is strictly decreasing on \((0, \infty)\). Therefore, \(R(x)\) is strictly decreasing on \((0, \infty)\) follows from Lemma 2.2 and (3.3) together with the monotonicity of the function \(f^{\prime}_{1}(x)/g^{\prime}_{1}(x)\) on the interval \((0, \infty)\) and \(f_{1}(\infty)=g_{1}(\infty)=0\).
Case 3: \(1/p< a\leq\Gamma^{p}(1+1/p)\). Then (3.1), (3.2), and Lemma 2.6 lead to
Note that (3.7) can be rewritten as
with \(u_{0}=1+(2-a)/[2(ap-1)]\in(1, \infty)\).
From (3.11) we clearly see that \(h_{1}(u)\) is strictly decreasing on \((1, u_{0})\) and strictly increasing on \((u_{0}, \infty)\). Then from (3.4), (3.6), and (3.9) we know that there exists \(\lambda\in(1, \infty)\) such that \(h(u)<0\) for \(u\in(1, \lambda)\) and \(h(u)>0\) for \(u\in(\lambda, \infty)\).
From (2.19) we clearly see that the function \(x\rightarrow u(x)\) is strictly decreasing from \((0, \infty)\) onto \((1, \infty)\). Then (3.5) and \(h(u)<0\) for \(u\in(1, \lambda)\) and \(h(u)>0\) for \(u\in (\lambda, \infty)\) lead to the conclusion that \(f^{\prime}_{1}(x)/g^{\prime}_{1}(x)\) is strictly increasing on \((0, \mu)\) and strictly decreasing on \((\mu, \infty)\), where \(\mu=[a/(\lambda^{p}-1)]^{1/p}\).
Therefore, \(R(x)\) is strictly decreasing on \((0, \infty)\) follows from (3.3), (3.10), Lemma 2.1(1), and the piecewise monotonicity of the function \(f^{\prime}_{1}(x)/g^{\prime}_{1}(x)\) on the interval \((0, \infty)\) together with the fact that \(g^{\prime}_{1}(x)=-e^{-x^{p}}<0\) and \(f_{1}(\infty)=g_{1}(\infty)=0\).
Case 4: \(\Gamma^{p}(1+1/p)< a<2\). Then we clearly see that (3.9) and (3.11) again hold. Making use of the same method as in Case 3 we know that there exists \(\eta>0\) such that \(f^{\prime}_{1}(x)/g^{\prime}_{1}(x)\) is strictly increasing on \((0, \eta)\) and strictly decreasing on \((\eta, \infty)\).
It follows from Lemma 2.6 that
Therefore, there exists \(x_{0}\in(0, \infty)\) such that \(R(x)\) is strictly increasing on \((0, x_{0})\) and strictly decreasing on \((x_{0}, \infty)\) follows from (3.3), (3.12), Lemma 2.1(2), and the piecewise monotonicity of the function \(f^{\prime}_{1}(x)/g^{\prime}_{1}(x)\) on the interval \((0, \infty)\) together with the fact that \(g^{\prime}_{1}(x)=-e^{-x^{p}}<0\) and \(f_{1}(\infty)=g_{1}(\infty)=0\). □
Let \(p>1\), \(x>0\), \(a>0\), \(R(x)\), \(f_{1}(x)\), \(g_{1}(x)\) and \(u=u(x)\) be defined by (1.5), (2.18), and (2.19), respectively. Then we clearly see that
It follows from (2.20), (2.21), (3.3), and (3.13) that
From (3.14) and (3.15) together with Theorem 3.1 we get Corollary 3.2 immediately.
Corollary 3.2
Let \(p>1\), \(a, x>0\), \(I_{p}(x)\) and \(R(x)\) be defined by (1.2) and (1.5), and \(x_{0}\) be the unique solution of the equation \(R^{\prime}(x)=0\) on the interval \((0, \infty)\) for \(\Gamma^{p}(1+1/p)< a<2\). Then the following statements are true:
-
(1)
if \(a\geq2\), then the double inequality
$$ \frac{1}{a} \bigl[ \bigl(x^{p}+a \bigr)^{1/p}-x \bigr]< I_{p}(x)< a^{-1/p}\Gamma \biggl(1+\frac{1}{p} \biggr) \bigl[ \bigl(x^{p}+a \bigr)^{1/p}-x \bigr] $$holds for all \(p>1\) and \(x>0\);
-
(2)
if \(0< a\leq\Gamma^{p}(1+1/p)\), then the double inequality
$$ a^{-1/p}\Gamma \biggl(1+\frac{1}{p} \biggr) \bigl[ \bigl(x^{p}+a \bigr)^{1/p}-x \bigr]< I_{p}(x)< \frac{1}{a} \bigl[ \bigl(x^{p}+a \bigr)^{1/p}-x \bigr] $$holds for all \(p>1\) and \(x>0\);
-
(3)
if \(\Gamma^{p}(1+1/p)< a<2\), then the two-sided inequality
$$ \frac{1}{R(x_{0})} \bigl[ \bigl(x^{p}+a \bigr)^{1/p}-x \bigr] \leq I_{p}(x)< \max \biggl\{ \frac{1}{a}, \frac{\Gamma (1+\frac{1}{p} )}{a^{1/p}} \biggr\} \bigl[ \bigl(x^{p}+a \bigr)^{1/p}-x \bigr] $$is valid for all \(p>1\) and \(x>0\).
Theorem 3.3
Let \(p>1\), \(a, b, x>0\), \(I_{p}(x)\) and \(a_{0}\) be defined by (1.2) and (1.4), respectively. Then the bilateral inequality
holds for all \(p>1\) and \(x>0\) if and only if \(a\geq2\) and \(b\leq a_{0}\).
Proof
If \(a\geq2\) and \(b\leq a_{0}\), then inequality (3.16) is valid for all \(p>1\) and \(x>0\) follows easily from (1.3) and Lemma 2.5.
If the inequality \(I_{p}(x)< [ (x^{p}+b )^{1/p}-x ]/b\) takes place for \(p>1\) and \(x>0\), then (3.14) leads to
which implies \(b\leq a_{0}\).
Next, we use the proof by contradiction to prove that \(a\geq2\) if the inequality \(I_{p}(x)> [ (x^{p}+b )^{1/p}-x ]/a\) holds for all \(x>0\) and \(p>1\).
From Lemmas 2.3 and 2.4 we clearly see that
We divide the proof into two cases.
Case 1: \(a\leq a_{0}\). Then it follows from the sufficiency of Theorem 3.3 which was proved previously that \(I_{p}(x)< [ (x^{p}+b )^{1/p}-x ]/a\) for all \(p>1\) and \(x>0\).
Case 2: \(a_{0}< a<2\). Let \(R(x)\) be defined by (1.5), then Theorem 3.1(3), (3.15), and (3.17) lead to the conclusion that there exists \(x_{0}\in(0, \infty)\) such that \(R(x)\) is strictly decreasing on \((x_{0}, \infty)\) and
or
for all \(p>1\) and \(x\in(x_{0}, \infty)\). □
Let \(p>1\), \(a>0\), \(x>0\), \(q=1/p\in(0, 1)\), and \(u=x^{p}>0\). Then from (1.1) and (1.2) one has
and Corollary 3.2 and Theorem 3.3 can be rewritten as follows.
Corollary 3.4
Let \(q\in(0, 1)\), \(a>0\), and \(u>0\). Then the following statements are true:
-
(1)
if \(a\geq2\), then the double inequality
$$ \frac{(u+a)^{q}-u^{q}}{qa}< e^{u}\Gamma(q,u)< \frac{\Gamma(1+q) [(u+a)^{q}-u^{q} ]}{qa^{q}} $$(3.18)holds for all \(q\in(0, 1)\) and \(u>0\), and inequality (3.18) is reversed if \(0< a\leq\Gamma^{1/q}(1+q)\);
-
(2)
if \(\Gamma^{1/q}(1+q)< a<2\), then the two-sided inequality
$$ \frac{(u+a)^{q}-u^{q}}{q\theta(q, u_{0}, a)}\leq e^{u}\Gamma(q,u)< \max \biggl\{ \frac{1}{a}, \frac{\Gamma(1+q)}{a^{q}} \biggr\} \frac{(u+a)^{q}-u^{q}}{q} $$holds for all \(q\in(0, 1)\) and \(u>0\), where \(\theta(q, u_{0}, a)= [(u_{0}+a)^{q}-u^{q}_{0} ]/ [qe^{u_{0}}\Gamma(q, u_{0}) ]\) and \(u_{0}\) is the unique solution of the equation
$$ \frac{d [ \frac{(u+a)^{q}-u^{q}}{qe^{u}\Gamma(q,u)} ]}{du}=0 $$on the interval \((0, \infty)\) for \(\Gamma^{1/q}(1+q)< a<2\).
Corollary 3.5
Let \(a, b, u>0\), \(q\in(0, 1)\) and \(a_{0}\) be defined by (1.4). Then the double inequality
holds for all \(q\in(0, 1)\) and \(u>0\) if and only if \(a\geq2\) and \(b\leq a_{0}\).
Let \(q\rightarrow0^{+}\) and \(Ei(u)=\lim_{q\rightarrow 0^{+}}\Gamma(q, u)\). Then Corollaries 3.4 and 3.5 lead to Remarks 3.6 and 3.7.
Remark 3.6
Let \(a>0\) and \(u>0\), then the following statements are true:
-
(1)
if \(a\geq2\), then the double inequality
$$ \frac{\log (1+\frac{a}{u} )}{a}< e^{u}Ei(u)< \log \biggl(1+ \frac {a}{u} \biggr) $$(3.19)holds for all \(u>0\), and inequality (3.19) is reversed if \(0< a< e^{-\gamma}\);
-
(2)
if \(e^{-\gamma}< a<2\), then we have the sided inequality
$$ \frac{e^{u_{0}}Ei(u_{0})}{\log (1+\frac{a}{u_{0}} )}\log \biggl(1+\frac{a}{u} \biggr)\leq e^{u}Ei(u)< \max \biggl\{ \frac{1}{a}, 1 \biggr\} \log \biggl(1+ \frac{a}{u} \biggr) $$(3.20)for all \(u>0\), where \(u_{0}\) is the unique solution of the equation
$$ \frac{d}{du}\frac{\log (1+\frac{a}{u} )}{e^{u}Ei(u)}=0 $$(3.21)on the interval \((0, \infty)\) for \(e^{-\gamma}< a<2\).
Remark 3.7
Let \(a, b>0\) and \(a_{0}\) be defined by (1.4). Then the double inequality
holds for all \(u>0\) if and only if \(a\geq2\) and \(b\leq a_{0}\).
In particular, if \(a=1\), then numerical computations show that \(u_{0}=0.23855\ldots\) is the unique solution of the equation
and \(e^{u_{0}}Ei(u_{0})/\log(1+1/u_{0})=0.83311\ldots>8\text{,}331/10\text{,}000\). Therefore, Remark 3.7 leads to Remark 3.8.
Remark 3.8
The double inequality
is valid for all \(u>0\).
Remark 3.9
Unfortunately, in the article we cannot deal with the monotonicity for the function \(R(x)\) defined by (1.5) and present the bounds for the function \(I_{p}(x)\) given by (1.2) in the case of \(p\in(0,1)\); we leave it as an open problem to the reader.
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Acknowledgements
The research was supported by the Natural Science Foundation of China under Grants 61374086, 11371125, and 11401191.
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Yang, ZH., Zhang, W. & Chu, YM. Monotonicity and inequalities involving the incomplete gamma function. J Inequal Appl 2016, 221 (2016). https://doi.org/10.1186/s13660-016-1160-7
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DOI: https://doi.org/10.1186/s13660-016-1160-7