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Refinement of integral inequalities for monotone functions
Journal of Inequalities and Applications volume 2012, Article number: 301 (2012)
Abstract
In this paper, we give refinements of some inequalities for generalized monotone functions by using logconvexity of some functionals.
1 Introduction
Let us denote
We consider the following theorem of Heinig and Maligranda.
Theorem 1.1 [1]
Let \mathrm{\infty}\le a<b\le \mathrm{\infty} and let f and g be positive functions on (a,b), where g is continuous on (a,b).

(a)
Suppose that f is a decreasing function on (a,b) and g is an increasing function on (a,b), where g(a+0)=0. Then, for any p\in (0,1],
{H}_{1}(f,g)\le {H}_{p}(f,g).(1)
If 1\le p<\mathrm{\infty}, then the inequality (1) holds in the reversed direction.

(b)
Suppose that f is an increasing function on (a,b) and g is a decreasing function on (a,b), where g(b0)=0. Then, for any p\in (0,1],
{\tilde{H}}_{1}(f,g)\le {\tilde{H}}_{p}(f,g).(2)
If 1\le p<\mathrm{\infty}, then the inequality (2) holds in the reversed direction.
We consider positive real valued functions f, g defined on an interval (a,b), \mathrm{\infty}\le a<b\le \mathrm{\infty}. We say that f is Cdecreasing (Cincreasing), C\ge 1, if f(x)\le Cf(y) (f(y)\le Cf(x)) whenever y\le x, y,x\in (a,b).
Now, throughout the paper, f is nonnegative and g is a positive function. Some extensions of Theorem 1.1 were obtained in [2] as follows.
Theorem 1.2 [2]
Assume that 0<p<q<\mathrm{\infty} and \mathrm{\infty}\le a<b\le \mathrm{\infty}.

(a)
If f is Cdecreasing and g is increasing and differentiable such that g(a+0)=0, then
{H}_{q}(f,g)\le {C}^{1\frac{p}{q}}{H}_{p}(f,g).(3) 
(b)
If f is Cincreasing and g is increasing and differentiable such that g(a+0)=0, then
{H}_{q}(f,g)\ge {C}^{\frac{p}{q}1}{H}_{p}(f,g).(4) 
(c)
If f is Cincreasing and g is decreasing and differentiable such that g(b0)=0, then
{\tilde{H}}_{q}(f,g)\le {C}^{1\frac{p}{q}}{\tilde{H}}_{p}(f,g).(5) 
(d)
If f is Cdecreasing and g is decreasing and differentiable such that g(b0)=0, then
{\tilde{H}}_{q}(f,g)\ge {C}^{\frac{p}{q}1}{\tilde{H}}_{p}(f,g).(6)
As a special case, we consider Cmonotone functions with respect to power functions.
For {C}_{1},{C}_{2}\ge 1, \mathrm{\infty}<{\alpha}_{1}\le {\alpha}_{2}<\mathrm{\infty}, we say that f\in {Q}^{{\alpha}_{1}}({C}_{1}) if f(x){x}^{{\alpha}_{1}} is {C}_{1}increasing and f\in {Q}_{{\alpha}_{2}}({C}_{2}) if f(x){x}^{{\alpha}_{2}} is {C}_{2}decreasing.
Theorem 1.3 [2]Let 0<p\le q<\mathrm{\infty}.

(a)
If f\in {Q}^{{\alpha}_{1}}(C), \alpha >{\alpha}_{1}, then for any x\ge 0,
{G}_{q}(f,x)\le {p}^{1/p}{q}^{1/q}{(\alpha {\alpha}_{1})}^{1/p1/q}{C}^{1p/q}{G}_{p}(f,x).(7) 
(b)
If f\in {Q}_{{\alpha}_{2}}(C), {\alpha}_{2}>\alpha, then for any x\ge 0,
{\tilde{G}}_{q}(f,x)\le {p}^{1/p}{q}^{1/q}{({\alpha}_{2}\alpha )}^{1/p1/q}{C}^{1p/q}{\tilde{G}}_{p}(f,x).(8)
2 Main results
In this paper, we prove some improvements and refinements of the above results by using the logconvexity method [3]. We consider the following theorem.
Theorem 2.1 Let \varphi :[0,\mathrm{\infty})\to \mathbb{R} be a convex and differentiable function such that \varphi (0)=0 and let \mathrm{\infty}\le a<b\le \mathrm{\infty}.

(a)
If f is Cdecreasing and g is increasing and differentiable such that g(a+0)=0, then
\varphi (C{\int}_{a}^{b}f(x)\phantom{\rule{0.2em}{0ex}}dg(x))\ge C{\int}_{a}^{b}{\varphi}^{\prime}(f(x)g(x))f(x)\phantom{\rule{0.2em}{0ex}}dg(x).(9) 
(b)
If f is Cincreasing and g is increasing and differentiable such that g(a+0)=0, then
\varphi (\frac{1}{C}{\int}_{a}^{b}f(x)\phantom{\rule{0.2em}{0ex}}dg(x))\le \frac{1}{C}{\int}_{a}^{b}{\varphi}^{\prime}(f(x)g(x))f(x)\phantom{\rule{0.2em}{0ex}}dg(x).(10) 
(c)
If f is Cincreasing and g is decreasing and differentiable such that g(b0)=0, then
\varphi (C{\int}_{a}^{b}f(x)\phantom{\rule{0.2em}{0ex}}d[g(x)])\ge C{\int}_{a}^{b}{\varphi}^{\prime}(f(x)g(x))f(x)\phantom{\rule{0.2em}{0ex}}d[g(x)].(11) 
(d)
If f is Cdecreasing and g is decreasing and differentiable such that g(b0)=0, then
\varphi (\frac{1}{C}{\int}_{a}^{b}f(x)\phantom{\rule{0.2em}{0ex}}d[g(x)])\le \frac{1}{C}{\int}_{a}^{b}{\varphi}^{\prime}(f(x)g(x))f(x)\phantom{\rule{0.2em}{0ex}}d[g(x)].(12) 
(e)
If the condition ‘ϕ is convex’ is replaced by ‘ϕ is concave,’ then all the inequalities (9)(12) hold in the reversed direction.
Remark 2.2 It was given in [2] that ϕ is a nonnegative convex function, but from the proof of Theorem 2.1 given there, it is clear that the results are still valid without the condition of nonnegativity of ϕ.
Remark 2.3 For the special case \varphi (x)={x}^{p}, p>1, the formulas (9)(12) are as follows:
and
If the condition p>1 is replaced by 0<p<1, then all the inequalities (13)(16) hold in the reversed direction.
We consider the following functionals.
(M_{1}) Under the assumptions of Theorem 2.1(a), we define a linear functional as
(M_{2}) Under the assumptions of Theorem 2.1(b), we define a linear functional as
(M_{3}) Under the assumptions of Theorem 2.1(c), we define a linear functional as
(M_{4}) Under the assumptions of Theorem 2.1(d), we define a linear functional as
Remark 2.4 Under the assumptions of Theorem 2.1 with ϕ as a convex function, the linear functionals {\mathcal{L}}_{i}(\varphi )\ge 0 for i=1,\dots ,4.
We will consider the classical method from [3] (see also [4] and the references given in it) to prove the logconvexity of the functionals defined as above by considering a convex function defined in the following lemma.
Lemma 2.5 Let a family of functions {\varphi}_{p}:[0,\mathrm{\infty})\to \mathbb{R}, p>0, be defined by
with 0log0=0. Then {\varphi}_{p}^{\u2033}(x)={x}^{p2}, that is, {\varphi}_{p} is convex for x>0.
Let us denote
and
Using functions defined in Lemma 2.5, we get
We will prove the logconvexity and related results for functionals {\mathcal{L}}_{i}, i=1,\dots ,4.
Theorem 2.6 Let linear functionals {\mathcal{L}}_{i}, i=1,\dots ,4 be defined as above and {\mathcal{L}}_{i}({\varphi}_{p}) be positive. Then for i=1,\dots ,4,

(a)
for all p,q>0
{\mathcal{L}}_{i}^{2}({\varphi}_{\frac{p+q}{2}})\le {\mathcal{L}}_{i}({\varphi}_{p}){\mathcal{L}}_{i}({\varphi}_{q}),(22)
that is, p\mapsto {\mathcal{L}}_{i}({\varphi}_{p}) is logconvex in the Jensen sense;

(b)
also, p\mapsto {\mathcal{L}}_{i}({\varphi}_{p}) is logconvex; that is, for p<q<r (p,q,r\in {\mathbb{R}}^{+})
{({\mathcal{L}}_{i}({\varphi}_{q}))}^{rp}\le {({\mathcal{L}}_{i}({\varphi}_{p}))}^{rq}{({\mathcal{L}}_{i}({\varphi}_{r}))}^{qp}.(23)
Proof (a) Suppose that i=1,\dots ,4 is arbitrary.
We shall use the idea from [[3], Theorem 4]. Let us consider the function defined by
where r=\frac{p+q}{2}, u,w\in \mathbb{R}. We have
Therefore, λ is convex for x>0. Hence, {\mathcal{L}}_{i}(\lambda )\ge 0, that is,
and therefore we get (22).

(b)
Since {\mathcal{L}}_{i} is continuous, so it is logconvex. Therefore, (23) is valid too.
Since i was taken to be arbitrary, so the above results hold for all i=1,\dots ,4. □
Corollary 2.7 If s>0, p<q<r (p,q,r\in {\mathbb{R}}^{+}) and p,q,r\ne s, then the following inequalities hold:
Proof For i=1, we have
Since s>0, so p/s<q/s<r/s. Also, for f is Cdecreasing, {f}^{s} is {C}^{s}decreasing. We make substitutions f\to {f}^{s}, g\to {g}^{s}, C\to {C}^{s}, p\to p/s, q\to q/s, and r\to r/s in (23). We get
After simplification, we get (24). Similarly, for i=2,3,4, we get (25)(27) respectively. □
Remark 2.8 From the inequalities (24)(27) for (q<s), we get the refinement for inequalities obtained from Theorem 1.2 and reversion when (q>s). Of course, we can get such refinement and reversions in all other cases for p, s and r, s.
Corollary 2.9 For s>0, p<q<r (p,q,r\in {\mathbb{R}}^{+}) and p,q,r\ne s.

(a)
If f\in {Q}^{{\alpha}_{1}}(C), \alpha >{\alpha}_{1}, then for any x>0, the following inequality holds:
(28) 
(b)
If f\in {Q}_{{\alpha}_{2}}(C), {\alpha}_{2}>\alpha, then for any x\ge 0, the following inequality holds:
(29)
Proof (a) It is a simple consequence of Corollary 2.7. Since f\in {Q}^{{\alpha}_{1}}(C), by making substitutions f\to f(t){t}^{{\alpha}_{1}} and g\to {t}^{({\alpha}_{1}\alpha )} in (26), we get (28).

(b)
Since f\in {Q}^{{\alpha}_{2}}(C), by making substitutions f\to f(t){t}^{{\alpha}_{2}} and g\to {t}^{({\alpha}_{2}\alpha )} in (24), we get (29). □
Now, we state and prove the Lagrangetype mean value theorem for the linear functionals {\mathcal{L}}_{i}, i=1,\dots ,4 defined by (M_{1})(M_{4}).
Theorem 2.10 Let {\mathcal{L}}_{i}, i=1,\dots ,4 be linear functionals defined by (M_{1})(M_{4}) and \varphi \in {C}^{2}[0,a], a>0, such that \varphi (0)=0. Then there exists {\xi}_{i}\in [0,a] such that the identity
holds for i=1,\dots ,4.
Proof Fix i=1,\dots ,4.
Since {\varphi}^{\u2033} is continuous on [0,a], it attains its maximum and minimum value on [0,a]. Let
Let us consider functions {F}_{1},{F}_{2}:[0,a]\to \mathbb{R} defined by
Clearly,
and
so {F}_{1}, {F}_{2} are convex functions. Also, {F}_{1}(0)=0={F}_{2}(0). Hence, from Theorem 2.1 for {F}_{1} and {F}_{2} respectively, it follows
and
Combining (31) and (32), we get
If {\mathcal{L}}_{i}({x}^{2})=0, then {\mathcal{L}}_{i}(\varphi )=0 and (30) holds for all {\xi}_{i}\in [0,a]. Otherwise,
Since {\varphi}^{\u2033} is continuous, there exists {\xi}_{i}\in [0,a] such that (30) holds and the proof is complete. □
Theorem 2.11 Let {\mathcal{L}}_{i}, i=1,\dots ,4 be linear functionals defined by (M_{1})(M_{4}) and \varphi ,\psi \in {C}^{2}[0,a], a>0, such that \varphi (0)=0=\psi (0). Then there exists {\xi}_{i}\in [0,a] such that the identity
holds for i=1,\dots ,4, provided that denominators are nonzero.
Proof Fix 1\le i\le 4 and define L\in {C}^{2}[0,a] in the way that
where {c}_{1} and {c}_{2} are defined by {c}_{1}={\mathcal{L}}_{i}(\psi ) and {c}_{2}={\mathcal{L}}_{i}(\varphi ). Now, from Theorem 2.10 for the function L, it follows
Since for (33) the denominators are nonzero, we have {\mathcal{L}}_{i}({x}^{2})\ne 0 (because if it is zero, then {\mathcal{L}}_{i}(\psi )=0 by Theorem 2.10). Therefore, (34) gives (33). □
Corollary 2.12 Let {\mathcal{L}}_{i}, i=1,\dots ,4 be linear functionals defined by (M_{1})(M_{4}). For distinct positive real numbers l and r different from one, there exists {\xi}_{i}\in [0,a] such that
holds for i=1,\dots ,4.
Proof Taking \varphi (x)={x}^{l} and \psi (x)={x}^{r} in (33), for distinct positive real numbers l and r different from one, we obtain (35). □
Remark 2.13 Since for fix i=1,\dots ,4 the function {\xi}_{i}\to {\xi}_{i}^{lr}, l\ne r is invertible, then from (35) we get
3 Cauchy means
In this section we deduce Cauchy means from Theorem 2.11. Suppose that {\varphi}^{\u2033}/{\psi}^{\u2033} has inverse. Then (33) gives
We conclude that the expression on the righthand side of the above equation is also a mean. For r,l\in {\mathbb{R}}^{+}, we define the Cauchy means
Also, we have continuous extensions of these means in other cases. Therefore, by limit, we have the following:
We also need the following result (see, e.g., [5]).
Lemma 3.1 If Φ is a convex function on an interval I\subset \mathbb{R} and if r\le u, l\le v, r\ne l, u\ne v, then the following inequality is valid:
Now, we deduce the monotonicity of means defined by (38) in the form of Dresher’s inequality as follows.
Theorem 3.2 Let {M}_{l,r}^{i} be given as in (38) and r,l,u,v\in {\mathbb{R}}^{+} be such that l\le v, r\le u. Then
Proof By Theorem 2.6, {\mathcal{L}}_{i} is logconvex. We set \mathrm{\Phi}(l)=log{\mathcal{L}}_{i}({\varphi}_{l}) in Lemma 3.1 and get
By using the properties of a log function, we get immediately (44). □
Corollary 3.3 For distinct positive real numbers l, r and s, there exist {\xi}_{i}\in [0,a], i=1,\dots ,4 such that the following identities hold:
Proof For i=1, making substitutions f\to {f}^{s}, g\to {g}^{s}, C\to {C}^{s}, \varphi (x)={x}^{l/s}, and \psi (x)={x}^{r/s} in (33), we get (46).
Similarly, for i=2,3,4, making substitutions as above in (33), we get (47), (48) and (49) respectively. □
Remark 3.4 Since the function {\xi}_{i}\to {\xi}_{i}^{lr} is invertible for all i=1,\dots ,4, from (46)(49), we can again formulate the corresponding Cauchy means for distinct positive real numbers l, r and s.
They are given as follows:
Corollary 3.5 Let {M}_{l,r,s}^{i}, i=1,\dots ,4 be given as above and r,l,u,v;s\in {\mathbb{R}}^{+} be such that l\le v, r\le u. Then
Proof By Theorem 3.2,
For s>0, we set f\to {f}^{s}, g\to {g}^{s}, C\to {C}^{s}, l\to l/s, r\to r/s, u\to v/s and r\to v/s in the above inequality for means and get (54). □
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Acknowledgements
The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. This research was partially funded by Higher Education Commission, Pakistan. The research of the second author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant 11711708890888.
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Butt, S.I., Pečarić, J. & Perić, I. Refinement of integral inequalities for monotone functions. J Inequal Appl 2012, 301 (2012). https://doi.org/10.1186/1029242X2012301
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DOI: https://doi.org/10.1186/1029242X2012301
Keywords
 convex function
 logconvex function
 Cauchy means
 mean value theorems