Abstract
Let \(\psi_{n}= ( -1 ) ^{n-1}\) \(\psi^{ ( n ) }\) (\(n=0,1,2,\ldots \)), where \(\psi^{ ( n ) }\) denotes the psi and polygamma functions. We prove that for \(n\geq0\) and two different real numbers a and b, the function
is strictly increasing from \(( -\min ( a,b ) ,\infty ) \) onto \(( \min ( a,b ) , ( a+b ) /2 ) \), which generalizes a well-known result. As an application, the complete monotonicity for a ratio of gamma functions is improved.
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1 Introduction
The classical Euler’s gamma and psi (or called digamma) functions are defined for \(x>0 \) by
respectively. Furthermore, the derivatives \(\psi^{\prime}, \psi ^{\prime\prime},\ldots, \psi^{ ( i ) }\) for \(i=1,2,\ldots \) , are called polygamma functions.
For convenience, we denote \(\psi_{n}(x)= ( -1 ) ^{n-1}\psi ^{ ( n ) }(x)\). It is well known that \(\psi_{n}(x)\) is strictly complete monotonic on \(( 0,\infty ) \); namely, \(( -1 ) ^{n-1}\psi^{ ( n ) } ( x ) >0\) for \(x>0\) and \(n\in\mathbb{N}\). Note that for the following integral and series representations (see [1], Sections 6.3, 6.4):
it is easy to see that \(\psi_{n} ( 0^{+} ) =\infty\) for \(n\geq0\), \(\psi_{n}( \infty) =0\) for \(n\geq1\), and \(\psi_{0} ( \infty ) =-\infty\). Moreover, \(\psi_{n}^{\prime}=-\psi_{ ( n+1 ) }(x)<0\).
Let \(f:I\rightarrow\mathbb{R}\) be strictly monotone and \(a,b\in I\). Then the so-called integral f-mean of a and b is defined in [2] by
For \(f=\psi\), Elezović and Pečarić [2], Theorem 6, proved an interesting result as follows.
Theorem EP
For \(x,a,b>0\), the digamma function ψ has the following properties:
-
(i)
\(I_{\psi^{\prime}} ( a,b ) \leq I_{\psi} ( a,b ) \); namely,
$$ \bigl( \psi^{\prime} \bigr) ^{-1} \biggl( \frac{\int_{a}^{b}\psi^{\prime } ( x )\,dx}{b-a} \biggr) \leq\psi^{-1} \biggl( \frac{ \int_{a}^{b}\psi ( x )\,dx}{b-a} \biggr). $$ -
(ii)
\(x\mapsto I_{\psi} ( x+a,x+b ) -x\) is increasing concave, and
$$ \lim_{x\rightarrow\infty} \bigl[ I_{\psi} ( x+a,x+b ) -x \bigr] = \frac{a+b}{2}. $$
Remark 1.1
It should be noted that, for \(a,b\in I\), if \(A ( a,b ) \) is a mean of a and b, then for \(x+a,x+b\in I\) the function \(x\mapsto A ( x+a,x+b ) -x\) is still a mean of a and b, which is due to the following relations:
Further, Batir [3], Theorem 2.7, gave a nice double inequality for \(I_{\psi_{n}} ( a,b ) \) as follows.
Theorem B
Let a and b be distinct positive real numbers and n be a positive integer. Then we have
or, equivalently,
where
is the generalized logarithmic mean of a and b.
An improvement of Theorem B was given in [4], Theorem 1, and [5], Theorem 1, by Qi as follows.
Theorem Q1
For real numbers \(a,b>0\) with \(a\neq b\) and an integer \(n\geq0\), the inequality
or
holds if \(p\leq-n\) and \(q\geq-n+1\), where \(S_{p} ( a,b ) \) is given in (1.3).
Motivated by the results just mentioned, the main aim of this paper is to continue the study of some further properties of the mean \(I_{\psi_{n}} ( a,b ) \) and \(I_{\psi _{n}} ( x+a,x+b ) -x\). More precisely, we have the following.
Theorem 1.2
For \(a,b>0\) with \(a\neq b\), the sequence \(\{I_{\psi _{n}} ( a,b ) \}_{n\geq0}\) is strictly decreasing, and
Theorem 1.3
Let a and b be distinct real numbers, and \(n\geq0\) be an integer. If \(\psi^{-1}_{n}\) is strictly decreasing with respected to x, then the function \(x\mapsto A_{\psi _{n}} ( x ) \) with
is strictly increasing from \(( -\min ( a,b ) ,\infty ) \) onto \(( \min ( a,b ) , ( a+b ) /2 ) \).
As a direct consequence, noting that \(\psi_{n}^{-1}\) is strictly decreasing, by Theorem 1.3 we have the following.
Corollary 1.4
Let a and b be distinct real numbers and \(n\geq0\) be an integer. Then for \(x>-\min ( a,b ) \) we have
where \(\min ( a,b ) \) and \(( a+b ) /2\) are the best constants. In particular, note that \(\psi_{0}=-\psi\), the double inequality
or
holds for \(x>-\min ( a,b ) \) with the best constants \(\min ( a,b ) \) and \(( a+b ) /2\).
Suppose that \(a,b>0\) with \(a\neq b\) in Theorem 1.3. Utilizing the strictly increasing property of \(x\mapsto A_{\psi_{n}} ( x ) \) on \(( 0,\infty ) \), we have \(A_{\psi_{n}} ( 0 ) < A_{\psi _{n}} ( x ) < A_{\psi_{n}} ( \infty ) \); namely,
Therefore, we conclude the following.
Corollary 1.5
Let \(a,b>0\) with \(a\neq b\) and \(n\geq0\) be an integer. Then for \(x>0\) we have
where \(I_{\psi_{n}} ( a,b ) \) and \(( a+b ) /2\) are the best constants. Particularly, noting that \(\psi_{0}=-\psi\), the double inequality
or
holds for \(x>0\) with the best constants \(I_{\psi} ( a,b ) \) and \(( a+b ) /2\).
We would think it worth noticing that the double inequality (1.6) was first proved in [6] by Elezović et al.
Remark 1.6
The second Kershaw double inequality [7] states that
for \(s\in ( 0,1 ) \) and \(x\geq0\). Some of the refinements, extensions, and generalizations of the double inequality (1.7) can be found in Qi’s review paper [8] and the references therein. It seems that our double inequality (1.5) may be the best second Kershaw type inequality, since the ranges of a and b in (1.5) are arbitrary real numbers, and the lower and upper bounds are sharp.
As an application of Theorem 1.3, we use it to prove a necessary and sufficient condition for the functions \(x\mapsto F_{a,b,c} ( x ) \) defined by (3.1) and \(x\mapsto 1/F_{a,b,c} ( x ) \) to be logarithmically monotonic on \(( -\rho,\infty ) \) with \(\rho=\min ( a,b,c ) \), which improves a well-known result.
2 Proofs of main results
This section we devote to the proof of our main results. First of all, let us give the following assertion, which is an improvement of Theorem 4 in [2].
Lemma 2.1
Let \(f\in C^{ ( 2 ) } ( I ) \). If f is strictly monotone, then the mean function
is strictly increasing (decreasing) according to \(f^{\prime\prime }/f^{\prime}\) being strictly increasing (decreasing).
Proof
By the Jensen inequality we have
if \(f^{\prime}\circ f^{-1}\) is strictly convex (concave).
Differentiation yields
where \(u=f^{-1}(x)\). This shows that \(f^{\prime}\circ f^{-1}\) is strictly convex if and only if both f and \(f^{\prime\prime}/f^{\prime}\) are either increasing or decreasing, and concave if and only if one of f and \(f^{\prime\prime}/f^{\prime}\) is increasing, while the other is decreasing.
Case 1: Both f and \(f^{\prime\prime}/f^{\prime}\) are increasing. Then \(f^{\prime}>0\) and \(f^{\prime}\circ f^{-1}\) is convex, and it follows from (2.2) that
Case 2: f is decreasing and \(f^{\prime\prime}/f^{\prime}\) is increasing. Then \(f^{\prime}<0\) and \(f^{\prime}\circ f^{-1}\) is concave and by (2.2) we also have \(dA_{f} ( t ) /dt>0\).
Case 3: Both f and \(f^{\prime\prime}/f^{\prime}\) are decreasing. Then \(f^{\prime}<0\) and \(f^{\prime}\circ f^{-1}\) is convex. Similarly, we have \(dA_{f} ( t ) /dt<0\).
Case 4: f is increasing and \(f^{\prime\prime}/f^{\prime}\) is decreasing. Then \(f^{\prime}>0\) and \(f^{\prime}\circ f^{-1}\) is concave. Obviously, we see that \(dA_{f} ( t ) /dt<0\).
To sum up, if \(f^{\prime\prime}/f^{\prime}\) is increasing (decreasing), then so is \(A_{f}\), which completes the proof. □
The following lemma is useful for our main proof, which is a generalization of Lemma 1.4 in [3] and Lemma 4 in [9].
Lemma 2.2
Let \(A: ( 0,\infty ) \times ( 0,\infty ) \rightarrow ( 0,\infty ) \) be a differentiable one-order homogeneous mean. Then, for all \(x+t,y+t\in ( 0,\infty ) \), we have
where \(p=A_{x} ( 1,1 ) \in [ 0,1 ] \). In particular, if \(A ( x,y ) \) is symmetric with respect to x and y, then
Proof
Using homogeneity of \(A(x,y)\) and the L’Hospital rule yield
In addition, it follows from [10] that
Putting the above together, we get (2.3).
In particular, if A is symmetric, that is, \(A ( x,y ) =A ( y,x ) \), then we clearly see that \(A_{x} ( x,y ) =A_{y} ( y,x ) \), and so \(A_{x} ( x,x ) =A_{y} ( x,x ) \). It follows from (2.5) that \(A_{x} ( x,x ) =A_{y} ( x,x ) =1/2\), and then (2.4) holds. The proof is complete. □
Lemma 2.3
Let \(\psi_{n}= ( -1 ) ^{n-1}\psi^{ ( n ) }\) for \(n\in\mathbb{N}\). Then all the following statements are true, and mutually equivalent.
-
(i)
the sequence \(\{\psi_{n+1}/\psi_{n}\}_{n\in\mathbb{N}}\) is strictly increasing;
-
(ii)
the function \(x\mapsto\psi_{n+1} ( x ) /\psi _{n} ( x ) \) is strictly decreasing on \(( 0,\infty ) \);
-
(iii)
the function \(x\mapsto\psi_{n} ( x ) \) is log-convex on \(( 0,\infty ) \).
Proof
(i) It suffices to prove \(\psi_{n+2}/\psi_{n+1}>\psi_{n+1}/\psi _{n}\) for \(n\in\mathbb{N}\), which is equivalent to \(\psi_{n+2}\psi_{n}-\psi _{n+1}^{2}>0\). By virtue of the integral representation given in (1.2), we get
which proves assertion (i).
(ii) Note that \(\psi_{n}^{\prime}=-\psi_{n+1}\), we have
which implies that the second assertion is true.
(iii) Differentiation gives
which completes the proof. □
Now we are in a position to prove our main results.
Proof of Theorem 1.2
We first prove that the sequence \(\{I_{\psi_{n}} ( a,b ) \} _{n\geq 0}\) is strictly decreasing, which means that for \(n\geq0\) the inequality
holds for \(a,b>0\) with \(a\neq b\). By the Jensen inequality, it suffices to check that \(\psi_{n+1}\circ\psi_{n}^{-1}\) is convex on \(( 0,\infty ) \). In fact, by Lemma 2.3 we have
where \(u=\psi_{n}^{-1} ( x ) \). This means that \(\psi _{n+1}\circ \psi_{n}^{-1}\) is convex, which proves inequality (2.6).
Taking \(p=-n\) and \(q=-n+1\) in Theorem Q1 gives
Considering that \(\lim_{p\rightarrow-\infty}S_{p} ( a,b ) =\min ( a,b ) \) in [11], then we get
which completes the proof. □
Proof of Theorem 1.3
To prove \(x\mapsto A_{\psi_{n}} ( x ) \) is strictly increasing on \(( -\min ( a,b ) ,\infty ) \), by Lemma 2.1 it suffices to check that \(\psi_{n}^{\prime\prime}/\psi_{n}^{\prime}\) is strictly increasing on \(( 0,\infty ) \). In fact, since \(\psi _{n}^{\prime}=-\psi_{n+1}\) we see that \(\psi_{n}^{\prime\prime }/\psi _{n}^{\prime}=-\psi_{n+2}/\psi_{n+1}\) is strictly increasing on \(( 0,\infty ) \) by the second assertion of Lemma 2.3. Thus, the increasing property of \(A_{\psi_{n}}\) follows.
As mentioned in the introduction, we see that \(\psi_{n} ( 0^{+} ) =\infty\) for \(n\geq0\), and so \(\psi_{n}^{-1} ( \infty ) =0\). Note that the symmetry of a and b, without loss of generality we may assume that \(b>a\). Then we have
which implies
To obtain \(\lim_{x\rightarrow\infty}A_{\psi_{n}} ( x ) = ( a+b ) /2\), we use (2.7) to derive that
Note that the generalized logarithmic mean \(S_{p} ( x,y ) \) is homogeneous and symmetric, it follows from Lemma 2.2 that
Therefore, we conclude that \(\lim_{x\rightarrow\infty}A_{\psi _{n}} ( x ) = ( a+b ) /2\), which completes the proof. □
3 An application
A function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and \(( -1 ) ^{n} ( f ( x ) ) ^{ ( n ) }\geq0\) for \(x\in I\) and n ≥0 (see [12]). A positive function f is called logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and its logarithm lnf satisfies \(( -1 ) ^{n} ( \ln f ( x ) ) ^{ ( n ) }\geq0\) for all \(n\in\mathbb{N}\) on I (see [13]). For convenience, we denote the sets of the completely monotonic functions and the logarithmically completely monotonic functions on I by \(\mathcal{C} [ I ] \) and \(\mathcal{L} [ I ] \), respectively. Qi in [14], Theorem 1, [15], Theorem 1, investigated the logarithmically complete monotonicity of the functions
and \(x\mapsto1/F_{a,b,c} ( x ) \). Furthermore, he concluded the following result.
Theorem Q2
Let \(a,b\), and c be real numbers and \(\rho=\min ( a,b,c ) \). If \(\theta ( t ) \) is an implicit function defined by
on \(( -\infty,\infty ) \), then \(\theta ( t ) \) is decreasing and \(t\theta ( t ) <0\) for \(\theta ( t ) \neq t\). Moreover:
-
(1)
\(F_{a,b,c} ( x ) \in\mathcal{L} [ ( -\rho,\infty ) ] \) if
$$\begin{aligned} ( a,b,c ) \in& \{ c\geq a,c\geq b \} \cup \bigl\{ c\geq a,0\geq c-b\geq\theta ( c-a ) \bigr\} \\ &{}\cup \bigl\{ c\leq a,c-b\geq\theta ( c-a ) \bigr\} \backslash \{ a=b=c \} . \end{aligned}$$ -
(2)
\(1/F_{a,b,c} ( x ) \in\mathcal{L} [ ( -\rho,\infty ) ] \) if
$$\begin{aligned} ( a,b,c ) \in& \{ c\leq a,c\leq b \} \cup \bigl\{ c\geq a,c-b\leq\theta ( c-a ) \bigr\} \\ &{}\cup \bigl\{ c\leq a,0\leq c-b\leq\theta ( c-a ) \bigr\} \backslash \{ a=b=c \} . \end{aligned}$$
Later, Qi and Guo in [16], Theorem 1, [17], Theorem 1, proved another result concerning the logarithmically complete monotonicity of the functions \(x\mapsto F_{a,b,c} ( x ) \) and \(x\mapsto1/F_{a,b,c} ( x ) \) for \(x>-\min ( a,b,c ) \), where \(c=c ( a,b ) \) is a constant depending on a and b. More precisely, they showed the following.
Theorem QG
Let a and b be two real numbers with \(a\neq b\) and \(c ( a,b ) \) be a constant depending on a and b.
-
(1)
If \(c ( a,b ) \leq\min ( a,b ) \), then \(1/F_{a,b,c} ( x ) \in\mathcal{L} [ ( -c ( a,b ) ,\infty ) ] \).
-
(2)
\(F_{a,b,c} ( x ) \in\mathcal{L} [ ( -\min ( a,b ) ,\infty ) ] \) if and only if \(c ( a,b ) \geq ( a+b ) /2\).
We would like to remark that the result in Theorem Q2 is rather interesting but somewhat complicated. Theorem QG shows that c is a constant depending on a and b, and \(c ( a,b ) \leq\min ( a,b ) \) is only sufficient for \(1/F_{a,b,c} ( x ) \in\mathcal{L} [ ( -c ( a,b ) ,\infty ) ] \). Here, we apply Theorem 1.3 to deduce that c is a constant independent of a and b, and \(c\leq\min ( a,b ) \) is also necessary for \(1/F_{a,b,c} ( x ) \in\mathcal{L} [ ( -c ( a,b ) ,\infty ) ] \). This improved result can be restated as follows.
Theorem 3.1
Let \(a,b\), and c be real numbers, and \(\rho=\min ( a,b,c ) \). Then \(1/F_{a,b,c} ( x ) \in\mathcal{L} [ ( -\rho,\infty ) ] \) if and only if \(c\leq\min ( a,b ) \), while \(F_{a,b,c} ( x ) \in\mathcal{L} [ ( -\rho,\infty ) ] \) if and only if \(c\geq ( a+b ) /2\).
Proof
For \(a\neq b\), we have
and
where \(\psi_{n}= ( -1 ) ^{n-1}\psi^{ ( n ) }\) and \(A_{\psi_{n}} ( x ) \) is defined by (1.4).
Since \(\psi_{n}^{\prime}=-\psi^{ ( n+1 ) }<0\), \(( \psi _{n}^{-1} ) ^{\prime}<0\), which means that \(\psi_{n}^{-1}\) is strictly decreasing on \(( 0,\infty ) \). This yields
for \(x\in ( -\rho,\infty ) \). Therefore, we have
Theorem 1.3 tells that \(x\mapsto A_{\psi_{n}} ( x ) =I_{\psi_{n}} ( x+a,x+b ) -x\) is strictly increasing from \(( -\min ( a,b ) ,\infty ) \) onto \(( \min ( a,b ) , ( a+b ) /2 ) \), which implies that
and
It is obvious that these are also true for \(a=b\). This completes the proof. □
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This paper is partially supported by the National Natural Science Foundation of China with grant No. 11371050.
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Yang, ZH., Zheng, SZ. Monotonicity of a mean related to polygamma functions with an application. J Inequal Appl 2016, 216 (2016). https://doi.org/10.1186/s13660-016-1155-4
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DOI: https://doi.org/10.1186/s13660-016-1155-4