- Research
- Open access
- Published:
A reverse Hölder inequality for \(\alpha,\beta\)-symmetric integral and some related results
Journal of Inequalities and Applications volume 2015, Article number: 138 (2015)
Abstract
In this paper, we establish a reversed Hölder inequality via an \(\alpha,\beta\)-symmetric integral, which is defined as a linear combination of the α-forward and the β-backward integrals, and then we give some generalizations of the \(\alpha,\beta\)-symmetric integral Hölder inequality which is due to Brito da Cruz et al.; some related inequalities are also given.
1 Introduction
Let \(a_{k} \ge0\), \(b_{k} \ge0\) (\(k=1,2,\ldots,n\)), \(p>1\), \(1/p+1/q=1\). The classical Hölder inequality [1] is stated as follows:
Similarly, the integral version of the Hölder inequality [1] is
where \(f(x)>0\), \(g(x)>0\), \(p>1\), \(1/p+1/q=1\), and \(f(x)\) and \(g(x)\) are continuous real-valued functions on \([a,b]\).
If \(p=q=2\), then inequalities (1.1) and (1.2) reduce to the well-known Cauchy inequalities [2] of the discrete form and the continuous form, respectively.
The Hölder inequality and Cauchy inequality play an important role in many areas of pure and applied mathematics. A large number of generalizations, refinements, variations, and applications of these inequalities have been investigated in [3–18] and references therein. Recently, Brito da Cruz et al. in [19] gave a \(\alpha,\beta \)-symmetric integral Hölder inequality as follows.
Let \(f, g: {\mathrm{R}}\to {\mathrm{R}}\) and \(a, b\in{\mathrm{R}}\) with \(a < b\). If \(\vert f\vert\) and \(\vert g\vert\) are \(\alpha,\beta\)-symmetric integrable on \([a,b]\), \(p>1\) with \(q = p/(p-1)\). Then
with equality if and only if the functions \(|f|\) and \(|g|\) are proportional.
The aim of this work is to establish a reversed version of the \(\alpha , \beta\)-symmetric integral Hölder inequality and some generalizations of the \(\alpha, \beta\)-symmetric integral Hölder inequality. Moreover, the obtained results will be applied to establish the \(\alpha, \beta\)-symmetric integral reverse Minkowski inequality, Dresher inequality, and their corresponding reverse versions. This paper is organized as follows. In Section 2, we recall some basic definitions and properties of \(\alpha,\beta\)-symmetric integral, which can also be found in [19, 20]; in Section 3, we establish a \(\alpha, \beta\)-symmetric integral reverse Hölder inequality and give some generalizations of the \(\alpha, \beta\)-symmetric integral Hölder inequality, we apply the obtained results to establish the reverse Minkowski inequality, Dresher inequality, and its reverse form involving \(\alpha, \beta\)-symmetric integral, some extensions of the Minkowski and Dresher inequalities are also given; in Section 4, we establish some further generalizations and refinements of the \(\alpha, \beta\)-symmetric integral Hölder inequality; in Section 5, we present a subdividing of the \(\alpha, \beta\)-symmetric integral Hölder inequality.
2 Preliminaries
In the section, we recall some basic definitions and properties of \(\alpha,\beta\)-symmetric integral.
The α-forward and β-backward differences are defined as follows (see [19]):
where \(\alpha>0\), \(\beta>0\).
Definition 2.1
(see [19])
Assume that \(I \subseteq {\mathrm{R}}\) with \(\sup I=+\infty\) and let \(a, b\in I\) with \(a< b\). If \(f:I\to{\mathrm{R}}\) and \(\alpha>0\), then the α-forward integral of f is defined by
where \(\int_{x}^{+\infty} {f(t)\Delta_{\alpha}t} =\alpha \sum_{k=0}^{+\infty} {f(x+k\alpha)} \), provided the series converges at \(x = a\) and \(x = b\).
The α-forward integral has the following properties.
Proposition 2.1
(see [19])
Assume that \(f, g:I\to {\mathrm{R}}\) are α-forward integrable on \([a,b]\), \(c \in[a,b]\), \(k\in{\mathrm{R}}\), then:
-
1.
\(\int_{a}^{a} {f(t)\Delta_{\alpha}t} =0\);
-
2.
\(\int_{a}^{b} {f(t)\Delta_{\alpha}t} =\int_{a}^{c} {f(t)\Delta_{\alpha}t} +\int_{c}^{b} {f(t)\Delta_{\alpha}t} \), when the integrals exist;
-
3.
\(\int_{a}^{b} {f(t)\Delta_{\alpha}t} =-\int_{b}^{a} {f(t)\Delta _{\alpha}t} \);
-
4.
kf is α-forward integrable on \([a,b]\) and
$$\int_{a}^{b} {kf(t)\Delta_{\alpha}t} =k\int _{a}^{b} {f(t)\Delta_{\alpha}t} ; $$ -
5.
\(f +g \) is α-forward integrable on \([a,b]\) and
$$\int_{a}^{b} {(f+g) (t)\Delta_{\alpha}t} = \int_{a}^{b} {f(t)\Delta_{\alpha}t} +\int _{a}^{b} {g(t)\Delta_{\alpha}t} ; $$ -
6.
if \(f \equiv0\), then \(\int_{a}^{b} {f(t)\Delta_{\alpha}t} =0 \).
Proposition 2.2
(see [19])
Assume that \(f:I\to{\mathrm{R}}\) is α-forward integrable on \([a,b]\). Let \(g:I \to{\mathrm{R}}\) be a nonnegative α-forward integrable function on \([a,b]\). Then fg is α-forward integrable on \([a,b]\).
Proposition 2.3
(see [19])
Assume that \(f:I \to{\mathrm{R}}\) and let \(|f| \) be α-forward integrable on \([a,b]\). If \(p>1\), then \(\vert f\vert^{p}\) is also α-forward integrable on \([a,b]\).
Proposition 2.4
(see [19])
Assume that \(f, g:I \to {\mathrm{R}}\) are α-forward integrable on \([a,b]\) with \(b=a+k\alpha\) for some \(k\in{\mathrm{N}}_{0}\). We have:
-
1.
If \(f(t)\ge0\) for all \(t\in\{a+k\alpha:k\in{\mathrm{N}}_{0} \}\), then \(\int_{a}^{b} {f(t)\Delta_{\alpha}t} \ge0 \).
-
2.
If \(g(t)\ge f(t)\) for all \(t\in\{a+k\alpha:k\in{\mathrm{N}}_{0} \} \), then \(\int_{a}^{b} {g(t)\Delta_{\alpha}t} \ge\int_{a}^{b} {f(t)\Delta_{\alpha}t}\).
Proposition 2.5
(Fundamental theorem of the α-forward integral, see [19])
Assume that \(f:I \to{\mathrm{R}}\) is α-forward integrable over I. Let \(x\in I\) and define \(F(x)=\int_{a}^{x} {f(t)\Delta _{\alpha}t} \). Then \(\Delta_{\alpha}[F](x)=f(x)\). Conversely, \(\int_{a}^{b} {\Delta_{\alpha}[f](t)\Delta_{\alpha}t} =f(b)-f(a)\).
Proposition 2.6
(α-Forward integration by parts, see [19])
Assume that \(f, g:I \to{\mathrm{R}}\). Let fg and \(f\Delta_{\alpha}[g]\) be α-forward integrable on \([a,b]\). Then
Similarly, the β-backward integral is defined by Definition 2.2.
Definition 2.2
(see [19])
Assume that \(I\subseteq {\mathrm{R}}\) with \(\inf I=-\infty\) and let \(a, b \in I\) with \(a< b\). For \(f:I\to \mathrm{R}\) and \(\beta>0\), the β-backward integral of f is defined by
where \(\int_{-\infty}^{x} {f(t)\nabla_{\beta}t} =\beta \sum_{k=0}^{+\infty} {f(x-k\beta)} \), provided the series converges at \(x=a\) and \(x=b\).
The β-backward integral has similar results and properties to the α-forward integral. In particular, the β-backward integral is the inverse operator of \(\nabla_{\beta}\).
We recall the \(\alpha,\beta\)-symmetric integral which is defined as a linear combination of the α-forward and the β-backward integrals.
Definition 2.3
(see [19])
Assume that \(f:{\mathrm{R}}\to {\mathrm{R}}\) and \(a, b \in \mathrm{R} \) with \(a< b\). If f is α-forward and β-backward integrable on \([a,b]\), \(\alpha,\beta\ge0\) with \(\alpha+\beta>0\), then we define the \(\alpha, \beta\)-symmetric integral of f from a to b by
The function f is \(\alpha,\beta\)-symmetric integrable if it is \(\alpha ,\beta\)-symmetric integrable for all \(a,b\in{\mathrm{R}}\).
The \(\alpha, \beta\)-symmetric integral has the following properties.
Proposition 2.7
(see [19])
Suppose that \(f, g:{\mathrm{R}} \to{\mathrm{R}}\) are \(\alpha,\beta\)-symmetric integrable on \([a,b]\). Let \(c \in[a,b]\) and \(k \in \mathrm{R}\). Then:
-
1.
\(\int_{a}^{a} {f(t)\, d_{\alpha,\beta} t} =0\);
-
2.
\(\int_{a}^{b} {f(t)\, d_{\alpha,\beta} t} =\int_{a}^{c} {f(t)\, d_{\alpha ,\beta} t} +\int_{c}^{b} {f(t)\, d_{\alpha,\beta} t}\), when the integrals exist;
-
3.
\(\int_{a}^{b} {f(t)\, d_{\alpha,\beta} t} =-\int_{b}^{a} {f(t)\, d_{\alpha,\beta} t}\);
-
4.
kf is \(\alpha, \beta\)-symmetric integrable on \([a,b]\) and
$$\int_{a}^{b} {kf(t)\, d_{\alpha,\beta} t} =k\int _{a}^{b} {f(t)\, d_{\alpha,\beta} t} ; $$ -
5.
\(f +g\) is \(\alpha,\beta\)-symmetric integrable on \([a,b]\) and
$$\int_{a}^{b} {(f+g) (t)\, d_{\alpha,\beta} t} = \int_{a}^{b} {f(t)\, d_{\alpha,\beta } t} +\int _{a}^{b} {g(t)\, d_{\alpha,\beta} t} ; $$ -
6.
fg is \(\alpha,\beta\)-symmetric integrable on \([a,b]\) provided g is a nonnegative function.
Proposition 2.8
(see [19])
Suppose that \(f:{\mathrm{R}}\to{\mathrm{R}}\) and \(p> 1\). Let \(\vert f \vert\) be symmetric \(\alpha ,\beta\)-integrable on \([a,b]\), then \(\vert f\vert^{p}\) is also \(\alpha,\beta\)-symmetric integrable on \([a,b]\).
Proposition 2.9
(see [19])
Assume that \(f, g:{\mathrm{R}}\to{\mathrm{R}}\) are \(\alpha,\beta\)-symmetric integrable functions on \([a,b]\), \(A=\{ a+k\alpha :k\in{\mathrm{N}}_{0} \}\) and \(B=\{b-k\beta:k\in{\mathrm{N}}_{0} \}\). For \(b \in A\) and \(a\in B\), we have:
-
1.
if \(\vert f(t)\vert\le g(t)\) for all \(t \in A \cup B\), then
$$\biggl\vert {\int_{a}^{b} {f(t)\, d_{\alpha,\beta} t} } \biggr\vert \le\int_{a}^{b} {g(t)\, d_{\alpha,\beta} t} ; $$ -
2.
if \(f(t)\ge0\) for all \(t\in A\cup B\), then
$$\int_{a}^{b} {f(t)\, d_{\alpha,\beta} t} \ge0; $$ -
3.
if \(g(t)\ge f(t)\) for all \(t\in A \cup B\), then
$$\int_{a}^{b} {g(t)}\, d_{\alpha,\beta} t\ge\int _{a}^{b} {f(t)\, d_{\alpha,\beta } t} . $$
In Proposition 2.9 we assume that \(a, b\in{\mathrm{R}}\) with \(b\in A=\{ a+k\alpha :k\in{\mathrm{N}}_{0} \}\) and \(a\in B=\{b-k\beta:k\in{\mathrm{N}}_{0} \}\), where \(\alpha,\beta\in{\mathrm{R}}_{0}^{+}\), \(\alpha+\beta\ne0\).
Proposition 2.10
(Mean value theorem, see [19])
If \(f, g:{\mathrm{R}}\to {\mathrm{R}}\) are bounded and \(\alpha,\beta\)-symmetric integrable on \([a,b]\) with g nonnegative. Let m and M be the infimum and the supremum, respectively, of the function f. Then there exists a real number K satisfying the inequalities \(m\le K\le M\) such that
3 Main results
As before, let \(a, b\in{\mathrm{R}}\) with \(b\in A=\{a+k\alpha:k\in {\mathrm{N}}_{0} \}\) and \(a\in B=\{b-k\beta:k\in{\mathrm{N}}_{0} \}\), where \(\alpha ,\beta\in{\mathrm{R}}_{0}^{+} \), \(\alpha+\beta\ne0\).
Theorem 3.1
(Reverse Hölder inequality)
Let \(f, g:{\mathrm{R}}\to {\mathrm{R}}\) and \(a, b\in{\mathrm{R}}\) with \(a < b\). If \(\vert f\vert\) and \(\vert g\vert\) are \(\alpha,\beta\)-symmetric integrable on \([a,b]\), \(0< p<1\) (or \(p<0\)) with \(q = p/(p-1)\), then
with equality if and only if the functions \(|f|\) and \(|g|\) are proportional.
Proof
We assume that
and let
and
Since the two functions \(\xi(t)\) and \(\gamma(t)\) are symmetric \(\alpha,\beta\)-integrable on \([a,b]\), applying the following reverse Young inequality (see [21]):
with equality holding iff \(x=y\), we have
Therefore, we obtain the desired inequality. □
Combining (1.3) and (3.1), we have Corollary 3.1.
Corollary 3.1
Let \(f_{j}:{\mathrm{R}}\to{\mathrm{R}}\), \(p_{j} \in {\mathrm{R}}\), \(j=1,2,\ldots, m\), \(\sum_{j=1}^{m} {1/p_{j} } =1\). If \(\vert f_{j} \vert\) is \(\alpha, \beta\)-symmetric integrable on \([a,b]\), then:
-
(1)
For \(p_{j} >1\), we have
$$ \int_{a}^{b} {\prod_{j=1}^{m} {\bigl\vert {f_{j} (t)} \bigr\vert \,d_{\alpha ,\beta} t} } \le \prod_{j=1}^{m} { \biggl(\int _{a}^{b} {\bigl\vert {f_{j} (t)} \bigr\vert ^{p_{j} }\,d_{\alpha,\beta} t} \biggr)^{1/p_{j} }} . $$(3.2) -
(2)
For \(0< p_{1} <1\), \(p_{j} <0\), \(j=2,\ldots, m\), we have
$$ \int_{a}^{b} {\prod_{j=1}^{m} {\bigl\vert {f_{j} (t)} \bigr\vert \,d_{\alpha ,\beta} t} } \ge \prod_{j=1}^{m} { \biggl(\int _{a}^{b} {\bigl\vert {f_{j} (t)} \bigr\vert ^{p_{j} }\,d_{\alpha,\beta} t} \biggr)^{1/p_{j} }} . $$(3.3)
Theorem 3.2
(Reverse Minkowski inequality)
Let \(f, g:{\mathrm{R}}\to {\mathrm{R}}\) and \(a, b, p\in{\mathrm{R}}\) with \(a< b\) and \(0< p< 1\) (or \(p<0\)). If \(|f|\) and \(|g|\) are \(\alpha,\beta\)-symmetric integrable on \([a,b]\), then
with equality if and only if the functions \(|f|\) and \(|g|\) are proportional.
Proof
Let
By the \(\alpha,\beta\)-symmetric integral Hölder inequality [12], in view of \(0< p<1\), we have
By using (3.5), we immediately arrive at the Minkowski inequality and the theorem is completely proved. □
An improvement of inequality (3.4) and its corresponding reverse form are obtained in Theorem 3.3.
Theorem 3.3
Assume that \(f, g:{\mathrm{R}}\to{\mathrm{R}}\) and \(a, b\in{\mathrm{R}}\) with \(a < b\). Assume that \(\vert f \vert\) and \(\vert g \vert\) are \(\alpha, \beta\)-symmetric integrable on \([a,b]\), \(p>0\), \(s,t\in {\mathrm{R}}\backslash\{0\}\), and \(s\ne t\).
-
(1)
Suppose that \(p, s, t\in{\mathrm{R}}\) are different, such that \(s, t>1\) and \((s-t)/(p-t)>1\), then
$$\begin{aligned}& \int_{a}^{b} {\bigl\vert f(x)+g(x)\bigr\vert ^{p}\,d_{\alpha,\beta} x} \\ & \quad \le \biggl[ { \biggl( {\int_{a}^{b} {\bigl\vert f(x)\bigr\vert ^{s}\,d_{\alpha,\beta} x} } \biggr)^{\frac{1}{s}}+ \biggl( {\int_{a}^{b} {\bigl\vert g(x)\bigr\vert ^{s}\,d_{\alpha ,\beta} x} } \biggr)^{\frac{1}{s}}} \biggr]^{s(p-t)/(s-t)} \\ & \qquad {} \times \biggl[ { \biggl( {\int_{a}^{b} { \bigl\vert f(x)\bigr\vert ^{t}\,d_{\alpha,\beta} x} } \biggr)^{\frac{1}{t}}+ \biggl( {\int_{a}^{b} {\bigl\vert g(x)\bigr\vert ^{t}\,d_{\alpha ,\beta} x} } \biggr)^{\frac{1}{t}}} \biggr]^{t(s-p)/(s-t)}. \end{aligned}$$(3.6) -
(2)
Suppose that \(p, s, t\in{\mathrm{R}}\) are different, such that \(0< s, t<1\) and \((s-t)/(p-t)<1\), then
$$\begin{aligned}& \int_{a}^{b} {\bigl\vert f(x)+g(x)\bigr\vert ^{p}\,d_{\alpha,\beta} x} \\ & \quad \ge \biggl[ { \biggl( {\int_{a}^{b} {\bigl\vert f(x)\bigr\vert ^{s}\,d_{\alpha,\beta} x} } \biggr)^{\frac{1}{s}}+ \biggl( {\int_{a}^{b} {\bigl\vert g(x)\bigr\vert ^{s}\,d_{\alpha ,\beta} x} } \biggr)^{\frac{1}{s}}} \biggr]^{s(p-t)/(s-t)} \\ & \qquad {} \times \biggl[ { \biggl( {\int_{a}^{b} { \bigl\vert f(x)\bigr\vert ^{t}\,d_{\alpha,\beta} x} } \biggr)^{\frac{1}{t}}+ \biggl( {\int_{a}^{b} {\bigl\vert g(x)\bigr\vert ^{t}\,d_{\alpha ,\beta} x} } \biggr)^{\frac{1}{t}}} \biggr]^{t(s-p)/(s-t)}. \end{aligned}$$(3.7)
Proof
(1) From the assumption, we have \((s-t)/(p-t)>1\), and it is obvious that
From the Hölder inequality (see [19]) with indices \((s-t)/(p-t)\) and \((s-t)/(s-p)\), it follows that
On the other hand, from the Minkowski inequality (see [19]) for \(s>1\) and \(t>1\), respectively, we obtain
and
From (3.8), (3.9), and (3.10), the desired result is obtained.
(2) Based on the assumption, we have \((s-t)/(p-t)<1\) and in view of
by using inequality (3.1) with indices \((s-t)/(p-t)\) and \((s-t)/(s-p)\), we have
On the other hand, thanks to the Minkowski inequality (3.4) for the cases of \(0< s<1\) and \(0< t<1\),
and
It follows from (3.11), (3.12), and (3.13) that the desired result is obtained. □
Remark 3.1
-
(1)
For Theorem 3.3, for \(p>1\), letting \(s=p+\varepsilon\), \(t=p-\varepsilon\), when p, s, t are different, \(s, t>1\), and \((s-t)/(p-t)/2>1\), and letting \(\varepsilon\to0\), we obtain the result of [19].
-
(2)
For Theorem 3.3, for \(0< p<1\), letting \(s=p+\varepsilon\), \(t=p-\varepsilon\), when p, s, t are different, \(0< s,t<1\), and \(0<(s-t)/(p-t)/2<1\), and letting \(\varepsilon\to0\), we obtain (3.4).
From the Minkowski inequality [19] and the reverse Minkowski inequality involving \(\alpha,\beta\)-symmetric integral, we can deduce the following generalization.
Corollary 3.2
Let \(f_{j}:{\mathrm{R}}\to{\mathrm{R}}\), \(j=1,2,\ldots, m\). If \(\vert f_{j} \vert\) is \(\alpha, \beta\)-symmetric integrable on \([a,b]\), then:
-
(1)
For \(p>1\), we have
$$ \Biggl( {\int_{a}^{b} {\Biggl\vert {\sum _{j=1}^{m} {f_{j} (x)} } \Biggr\vert ^{p}\, d_{\alpha ,\beta} x} } \Biggr)^{1/p}\le\sum _{j=1}^{m} { \biggl( {\int _{a}^{b} {\bigl\vert {f_{j}(x)} \bigr\vert ^{p}\, d_{\alpha,\beta} x} } \biggr)^{1/p}} . $$(3.14) -
(2)
For \(0< p<1\), we have
$$ \Biggl( {\int_{a}^{b} {\Biggl\vert {\sum _{j=1}^{m} {f_{j} (x)} } \Biggr\vert ^{p}\, d_{\alpha ,\beta} x} } \Biggr)^{1/p}\ge\sum _{j=1}^{m} { \biggl( {\int _{a}^{b} {\bigl\vert {f_{j}(x)} \bigr\vert ^{p}\, d_{\alpha,\beta} x} } \biggr)^{1/p}} . $$(3.15)
Corollary 3.3 is an analog of Corollary 3.2.
Corollary 3.3
Let \(f_{j}:{\mathrm{R}}\to{\mathrm{R}}\), \(j=1,2,\ldots, m\). If \(\vert f_{j} \vert\) is \(\alpha, \beta\)-symmetric integrable on \([a,b]\), then:
-
(1)
For \(p>1\), we have
$$ \int_{a}^{b} { \Biggl( {\sum _{j=1}^{m} {\bigl\vert f_{j} (x)\bigr\vert } } \Biggr)^{p}\, d_{\alpha ,\beta } x} \geq\sum _{j=1}^{m} {\int_{a}^{b} {\bigl\vert {f_{j} (x)} \bigr\vert ^{p}\, d_{\alpha ,\beta} x} } . $$(3.16) -
(2)
For \(0< p<1\), we have
$$ \int_{a}^{b} { \Biggl( {\sum _{j=1}^{m} {\bigl\vert f_{j} (x)\bigr\vert } } \Biggr)^{p}\, d_{\alpha ,\beta } x} \leq\sum _{j=1}^{m} {\int_{a}^{b} {\bigl\vert {f_{j} (x)} \bigr\vert ^{p}\, d_{\alpha ,\beta} x} } . $$(3.17)
Proof
(1) For \(p>1\), let \(s=p\), \(r=1\), by the Jensen inequality [3], it follows that
from the above inequality, we obtain
by integrating the above inequality with respect to x, we obtain the desired result.
(2) For \(0< p<1\), let \(s=1\), \(r=p\), by the Jensen inequality [3], we have
it follows from the above inequality that
by integrating the above inequality with respect to x, the desired result is obtained. □
Theorem 3.4
(Dresher inequality)
Let \(f, g:{\mathrm{R}}\to {\mathrm{R}}\) and \(a, b\in {\mathrm{R}}\) with \(a< b\). If \(\vert f \vert\) and \(\vert g\vert\) are \(\alpha,\beta\)-symmetric integrable on \([a,b]\), \(0< r<1<p\), then
with equality if and only if the functions \(|f|\) and \(|g|\) are proportional.
Proof
Based on the \(\alpha,\beta\)-symmetric integral Hölder and Minkowski inequalities [19], we have
From Theorem 3.2, we get
From (3.5) and (3.20), we get (3.18). Hence, the theorem is completely proved. □
Corollary 3.4
Let \(f_{j}:{\mathrm{R}}\to{\mathrm{R}}\), \(0< r<1<p\), \(j=1,2,\ldots, m\). If \(\vert f_{j} \vert\) is \(\alpha, \beta \)-symmetric integrable on \([a,b]\), then
Theorem 3.5
(Reverse Dresher inequality)
Let \(f, g:{\mathrm{R}}\to {\mathrm{R}}\) and \(a,b\in{\mathrm{R}}\) with \(a< b\). If \(\vert f \vert\) and \(\vert g\vert\) are \(\alpha,\beta\)-symmetric integrable on \([a,b]\), \(p\le 0\le r\le1\), then
with equality if and only if the functions \(|f|\) and \(|g|\) are proportional.
Proof
Let \(\alpha_{1} \ge0\), \(\alpha_{2} \ge0\), \(\beta_{1} >0\), and \(\beta_{2} >0\), and \(-1<\lambda<0\), using the following Radon inequality (see [3]):
we have
with equality if and only if \((\alpha)\) and \((\beta)\) are proportional. Let
and set \(\lambda=\frac{r}{p-r}\). From (3.23)-(3.25), we have
Since \(-1<\lambda=\frac{r}{p-r}<0\), we may assume \(p<0<r\), and by Theorem 3.2 and \(0< r\le1\), we obtain, respectively,
with equality if and only if f and g are proportional, and
with equality if and only if \(|f|\) and \(|g|\) are proportional.
From the equality conditions for (3.23), (3.27), and (3.28), it follows that the sign of equality in (3.22) holds if and only if \(|f|\) and \(|g|\) are proportional.
From (3.26)-(3.28), we obtain the reverse Dresher inequality and the theorem is completely proved. □
Corollary 3.5
Let \(f_{j}:{\mathrm{R}}\to{\mathrm{R}}\), \(p\le 0\le r<1\), \(j=1,2,\ldots, m\). If \(\vert f_{j} \vert\) is \(\alpha,\beta\)-symmetric integrable on \([a,b]\), then
4 Some further generalizations of the Hölder inequality
Theorem 4.1
Suppose that \(p_{k} >0\), \(\alpha_{kj} \in{\mathrm{R}}\) (\(j=1,2,\ldots ,m\), \(k=1,2,\ldots,s\)), \(\sum_{k}^{s} {\frac{1}{p_{k} }} =1\), \(\sum_{k=1}^{s} {\alpha_{kj} } =0\), \(f_{j} :{\mathrm{R}}\to{\mathrm{R}}\). If \(\vert f_{j}\vert\) is \(\alpha,\beta\)-symmetric integrable on \([a,b]\), then:
-
(1)
For \(p_{k} >1\), we have
$$ \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta} t} \le\prod_{k=1}^{s} { \Biggl( {\int_{a}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+p_{k} \alpha_{kj} }} \, d_{\alpha,\beta} t} } \Biggr)^{1/p_{k} }}. $$(4.1) -
(2)
For \(0< p_{s} <1\), \(p_{k} <0\) (\(k=1,2,\ldots,s-1\)), we have
$$ \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta} t} \ge\prod_{k=1}^{s} { \Biggl( {\int_{a}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+p_{k} \alpha_{kj} }}\, d_{\alpha,\beta} t} } \Biggr)^{1/p_{k} }}. $$(4.2)
Proof
(1) Let
Applying the assumptions \(\sum_{k}^{s} {\frac{1}{p_{k} }} =1\) and \(\sum_{k=1}^{s} {\alpha_{kj} } =0\), it follows from a direct computation that
That is,
It is obvious that
From the Hölder inequality (3.2), it follows that
Substituting \(g_{k} (t)\) in (4.5), we have inequality (4.1) immediately.
(2) The proof of inequality (4.2) is similar to the proof of inequality (4.1); by (4.3), (4.4), and (3.3), we obtain
Substitution of \(g_{k} (t)\) in Eq. (4.6) gives inequality (4.2) immediately. □
Remark 4.1
Let \(s=m\), \(\alpha_{kj} =-1/p_{k} \) for \(j\ne k\), and \(\alpha_{kk} =1-1/p_{k} \). Then the inequalities (4.1) and (4.2) are respectively reduced to (3.2) and (3.3).
Many existing inequalities concerned with the Hölder inequality are special cases of the inequalities (4.1) and (4.2). For example, we have the following.
Corollary 4.1
Under the assumptions of Theorem 4.1, assume that \(s=m\), \(\alpha_{kj} =-t/p_{k} \) for \(j\ne k\), and \(\alpha_{kk} =t(1-1/p_{k} )\) with \(t\in \mathrm{R}\), then:
-
(1)
For \(p_{k} >1\), one obtains
$$ \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta} t} \le\prod_{k=1}^{m} { \Biggl( {\int_{a}^{b} { \Biggl( {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert } } \Biggr)^{1-t}\bigl(\bigl\vert f_{k} (t)\bigr\vert ^{p_{k} }\bigr)^{t}\, d_{\alpha ,\beta} t} } \Biggr)^{1/p_{k} }} . $$(4.7) -
(2)
For \(0< p_{m} <1\), \(p_{k} <0\) (\(k=1,2,\ldots,m-1\)), one obtains
$$ \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta} t} \ge\prod_{k=1}^{m} { \Biggl( {\int_{a}^{b} { \Biggl( {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert } } \Biggr)^{1-t}\bigl(\bigl\vert f_{k} (t)\bigr\vert ^{p_{k} }\bigr)^{t}\, d_{\alpha ,\beta} t} } \Biggr)^{1/p_{k} }} . $$(4.8)
Theorem 4.2
Suppose that \(p_{k} >0\), \(r\in{\mathrm{R}}\), \(\alpha_{kj} \in{\mathrm{R}}\) (\(j=1,2,\ldots,m\), \(k=1,2,\ldots,s\)), \(\sum_{k}^{s} {\frac{1}{p_{k} }} =r\), \(\sum_{k=1}^{s} {\alpha_{kj} } =0\), \(f_{j} :{\mathrm{R}}\to{\mathrm{R}}\). If \(\vert f_{j} \vert\) is \(\alpha, \beta\)-symmetric integrable on \([a,b]\), then:
-
(1)
For \(rp_{k} >1\), we have
$$ \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta} t} \le\prod_{k=1}^{s} { \Biggl( {\int_{a}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+rp_{k} \alpha_{kj} }}\, d_{\alpha,\beta} t} } \Biggr)^{1/rp_{k} }}. $$(4.9) -
(2)
For \(0< rp_{s} <1\), \(rp_{k} <0\) (\(k=1,2,\ldots,s-1\)), we have
$$ \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta} t} \ge\prod_{k=1}^{s} { \Biggl( {\int_{a}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+rp_{k} \alpha_{kj} }}\, d_{\alpha,\beta} t} } \Biggr)^{1/rp_{k} }}. $$(4.10)
Proof
(1) Since \(rp_{k} >1\) and \(\sum_{k}^{s} {\frac{1}{p_{k} }} =r\), it follows that \(\sum_{k}^{s} {\frac{1}{rp_{k} }} =1\). Then by (4.1), we immediately find inequality (4.9).
(2) From \(0< rp_{s} <1\), \(rp_{k} <0\), and \(\sum_{k}^{s} {\frac{1}{p_{k} }} =r\), it follows that \(\sum_{k}^{s} {\frac{1}{rp_{k} }} =1\). By (4.2), we immediately get inequality (4.10). This completes the proof. □
From Theorem 4.2, we establish Corollary 4.2, which is a generalization of Theorem 4.2.
Corollary 4.2
Under the assumptions of Theorem 4.2, assume that \(s=2\), \(p_{1} =p\), \(p_{2} =q\), \(\alpha_{1j} =-\alpha_{2j} =\alpha_{j} \), then:
-
(1)
For \(rp>1\), we get
$$\begin{aligned}& \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta } t} \\& \quad \le \Biggl( {\int_{a}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+rp\alpha _{j} }}\, d_{\alpha,\beta} t} } \Biggr)^{1/rp} \Biggl( { \int_{a}^{b} {\prod_{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1-rq\alpha_{j} }}\, d_{\alpha ,\beta} t} } \Biggr)^{1/rq}. \end{aligned}$$(4.11) -
(2)
For \(0< rp<1\), we get
$$\begin{aligned}& \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta } t} \\& \quad \ge \Biggl( {\int_{a}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+rp\alpha _{j} }} \, d_{\alpha,\beta} t} } \Biggr)^{1/rp} \Biggl( { \int_{a}^{b} {\prod_{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1-rq\alpha_{j} }} \, d_{\alpha ,\beta} t} } \Biggr)^{1/rq}. \end{aligned}$$(4.12)
Now we present a refinement of inequalities (4.9) and (4.10), respectively.
Theorem 4.3
Under the assumptions of Theorem 4.2:
-
(1)
For \(rp_{k} >1\), one has
$$ \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta} t} \le\varphi(c)\le\prod_{k=1}^{s} { \Biggl( {\int_{a}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+rp_{k} \alpha_{kj} }} \, d_{\alpha ,\beta} t} } \Biggr)^{1/rp_{k} }} , $$(4.13)where
$$\varphi(c)\equiv\int_{a}^{c} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert } \, d_{\alpha,\beta} t} +\prod_{k=1}^{s} { \Biggl( {\int_{c}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+rp_{k} \alpha_{kj} }} \, d_{\alpha ,\beta} t} } \Biggr)^{1/rp_{k} }} $$is a nonincreasing function with \(a\le c\le b\).
-
(2)
For \(0< rp_{s} <1\), \(rp_{k} <0\) (\(k=1,2,\ldots,s-1\)), one has
$$ \int_{a}^{b} {\Biggl\vert \prod _{j=1}^{m} {f_{j} (t)} \Biggr\vert \, d_{\alpha,\beta} t} \ge\varphi(c)\ge\prod_{k=1}^{s} { \Biggl( {\int_{a}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+rp_{k} \alpha_{kj} }} \, d_{\alpha ,\beta} t} } \Biggr)^{1/rp_{k} }} , $$(4.14)where
$$\varphi(c)\equiv\int_{a}^{c} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert } \, d_{\alpha,\beta} t} +\prod_{k=1}^{s} { \Biggl( {\int_{c}^{b} {\prod _{j=1}^{m} {\bigl\vert f_{j} (t)\bigr\vert ^{1+rp_{k} \alpha_{kj} }} \, d_{\alpha ,\beta} t} } \Biggr)^{1/rp_{k} }} $$is a nondecreasing function with \(a\le c\le b\).
Proof
(1) Let
By rearrangement and the assumptions of Theorem 4.2, it follows that
Then thanks to the Hölder inequality (3.2), we have
Therefore, we obtain the desired result.
(2) The proof of inequality (4.14) is similar to the proof of inequality (4.13). □
5 A subdividing of the Hölder inequality
Theorem 5.1
Let \(f, g:{\mathrm{R}}\to{\mathrm{R}}\) and \(a,b\in{\mathrm{R}}\) with \(a < b\). Assume that \(\vert f \vert\) and \(\vert g \vert\) are \(\alpha ,\beta\)-symmetric integrable on \([a,b]\), and \(s,t\in{\mathrm{R}}\), and let \(p=(s-t)/1-t\), \(p=(s-t)/(s-1)\).
-
(1)
If \(s<1<t\) or \(s>1>t\), then
$$\begin{aligned} \int_{a}^{b} {\bigl\vert f(x)g(x)\bigr\vert \, d_{\alpha,\beta} x} \le& \biggl( {\int_{a}^{b} { \bigl\vert f(x)\bigr\vert ^{sp}\, d_{\alpha,\beta} x} } \biggr)^{1/p^{2}} \biggl( {\int_{a}^{b} {\bigl\vert g(x)\bigr\vert ^{tq}\, d_{\alpha,\beta } x} } \biggr)^{1/q^{2}} \\ &{}\times \biggl( {\int_{a}^{b} {\bigl\vert f(x)\bigr\vert ^{tp}\, d_{\alpha,\beta} x} \int_{a}^{b} {\bigl\vert g(x)\bigr\vert ^{sq}\, d_{\alpha,\beta} x} } \biggr)^{1/pq} \end{aligned}$$(5.1)with equality if and only if \(f(x)\) and \(g(x)\) are proportional.
-
(2)
If \(s>t>1\) or \(s< t<1\); \(t>s>1\) or \(t< s<1\), then
$$\begin{aligned} \int_{a}^{b} {\bigl\vert f(x)g(x)\bigr\vert \, d_{\alpha,\beta} x} \geq& \biggl( {\int_{a}^{b} { \bigl\vert f(x)\bigr\vert ^{sp}\, d_{\alpha,\beta} x} } \biggr)^{1/p^{2}} \biggl( {\int_{a}^{b} {\bigl\vert g(x)\bigr\vert ^{tq}\, d_{\alpha,\beta } x} } \biggr)^{1/q^{2}} \\ &{}\times \biggl( {\int_{a}^{b} {\bigl\vert f(x)\bigr\vert ^{tp}\, d_{\alpha,\beta} x} \int_{a}^{b} {\bigl\vert g(x)\bigr\vert ^{sq}\, d_{\alpha,\beta} x} } \biggr)^{1/pq} \end{aligned}$$(5.2)with equality if and only if \(f(x)\) and \(g(x)\) are proportional.
Proof
(1) Set \(p=\frac{s-t}{1-t}\), and it follows from \(s<1<t\) or \(s>1>t\) that
From inequality (1.3) with indices \(\frac{s-t}{1-t}\) and \(\frac{s-t}{s-1}\), it follows that
with equality if and only if \((fg)^{s}\) and \((fg)^{t}\) are proportional.
On the other hand, by the Hölder inequality again, for \(p=\frac{s-t}{1-t}>1\), the following two inequalities are obtained:
with equality if and only if \(f^{s(s-t)/(1-t)}\) and \(g^{s(s-t)/(s-1)}\) are proportional, and
with equality if and only if \(f^{t(s-t)/(1-t)}\) and \(g^{t(s-t)/(s-1)}\) are proportional.
It follows from (5.3), (5.4), and (5.5) that the case (1) of Theorem 5.1 is proved.
(2) Let \(p=\frac{s-t}{1-t}\), and in view of \(s>t>1\) or \(s< t<1\), we have
and \(t>s>1\) or \(t< s<1\), we have \(0<\frac{s-t}{1-t}<1\), by inequality (3.1) with indices \(\frac{s-t}{1-t}\) and \(\frac{s-t}{s-1}\), we have
with equality if and only if \((fg)^{s}\) and \((fg)^{t}\) are proportional.
On the other hand, from the reverse Hölder inequality again for \(0< p=\frac{s-t}{1-t}<1\) or \(p=\frac{s-t}{1-t}<0\), we obtain the following two inequalities:
with equality if and only if \(f^{s(s-t)/(1-t)}\) and \(g^{s(s-t)/(s-1)}\) are proportional, and
with equality if and only if \(f^{t(s-t)/(1-t)}\) and \(g^{t(s-t)/(s-1)}\) are proportional.
From (5.6), (5.7), and (5.8), the proof of the case (2) of Theorem 5.1 is completed. □
References
Beckenbach, EF, Bellman, R: Inequalities. Springer, Berlin (1961)
Mitrinović, DS: Analytic Inequalities. Springer, New York (1970)
Hardy, G, Littlewood, JE, Pólya, G: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)
Yang, X: A generalization of Hölder inequality. J. Math. Anal. Appl. 247, 328-330 (2000)
Yang, X: Refinement of Hölder inequality and application to Ostrowski inequality. Appl. Math. Comput. 138, 455-461 (2003)
Yang, X: A note on Hölder inequality. Appl. Math. Comput. 134, 319-322 (2003)
Yang, X: Hölder’s inequality. Appl. Math. Lett. 16, 897-903 (2003)
Abramovich, S, Pečarić, JE, Varošanec, S: Sharpening Hölder’s and Popoviciu’s inequalities via functionals. Rocky Mt. J. Math. 34, 793-810 (2004)
Abramovich, S, Pečarić, JE, Varošanec, S: Continuous sharpening of Hölder’s and Minkowski’s inequalities. Math. Inequal. Appl. 8(2), 179-190 (2005)
Wu, S, Debnath, L: Generalizations of Aczél’s inequality and Popoviciu’s inequality. Indian J. Pure Appl. Math. 36(2), 49-62 (2005)
He, WS: Generalization of a sharp Hölder’s inequality and its application. J. Math. Anal. Appl. 332, 741-750 (2007)
Wu, S: A new sharpened and generalized version of Hölder’s inequality and its applications. Appl. Math. Comput. 197, 708-714 (2008)
Kwon, EG, Bae, EK: On a continuous form of Hölder inequality. J. Math. Anal. Appl. 343, 585-592 (2008)
Nikolova, L, Varošanec, S: Refinements of Hölder’s inequality derived from functions \(\psi_{p,q,\lambda} \) and \(\phi_{p,q,\lambda}\). Ann. Funct. Anal. 2(1), 72-83 (2011)
Xu, B, Wang, X-H, Wei, W, Wang, H: On reverse Hilbert-type inequalities. J. Inequal. Appl. 2014, 198 (2014)
Yang, B, Chen, Q: A more accurate half-discrete reverse Hilbert-type inequality with a non-homogeneous kernel. J. Inequal. Appl. 2014, 96 (2014)
Yang, B: Multidimensional discrete Hilbert-type inequalities, operators and compositions. In: Analytic Number Theory, Approximation Theory, and Special Functions, pp. 429-484. Springer, New York (2014)
Zhao, C-J, Cheung, W-S: On subdividing of Hölder’s inequality. Far East J. Math. Sci.: FJMS 60(1), 101-108 (2012)
Brito da Cruz, AMC, Martins, N, Torres, DFM: A symmetric Nörlund sum with application to inequalities. In: Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics, vol. 47, pp. 495-503 (2013)
Brito da Cruz, AMC, Martins, N, Torres, DFM: A symmetric quantum calculus. In: Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics, vol. 47, pp. 359-366 (2013)
Xing, J-S, Su, K-Q, Tao, P-F: The applications of Young inequality and Young inverse inequality. J. Zhoukou Norm. Univ. 24(2), 37-39 (2007) (in Chinese)
Acknowledgements
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. This paper was partially supported by the Key Laboratory for Mixed and Missing Data Statistics of the Education Department of Guangxi Province (No. GXMMSL201404), the Scientific Research Project of Guangxi Education Department (Nos. YB2014560 and KY2015YB468), and the Natural Science Foundation of Guangxi Province (No. 2013JJAA10097).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Chen, GS., Wei, CD. A reverse Hölder inequality for \(\alpha,\beta\)-symmetric integral and some related results. J Inequal Appl 2015, 138 (2015). https://doi.org/10.1186/s13660-015-0645-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0645-0