A note on geometrically convex functions

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R E S E A R C H

Open Access

A note on geometrically convex functions

Muhamet Emin Özdemir

1

_{, Çetin Yildiz}

1*

_{and Mustafa Gürbüz}

2

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} K. K. Education Faculty, Atatürk University, Campus, Erzurum, 25240, Turkey

Full list of author information is available at the end of the article

Abstract

In this paper, we establish several new inequalities for twice diﬀerentiable mappings that are connected with the celebrated Hermite-Hadamard integral inequality. MSC: Primary 26D15; secondary 26A51

Keywords: Hermite-Hadamard inequality; geometrically convex function; (

α

,m)-geometrically convex function

1 Historical background and introduction

The following double inequality is well known in the literature as Hadamard’s inequality: Letf:I⊆R→Rbe a convex function deﬁned on an intervalIof real numbers,a,b∈I

anda<b, we have

f

a+b



≤ 

b–a b

a

f(x)dx≤f(a) +f(b)

 . (.)

Both inequalities hold in the reversed direction iff is concave.

It was ﬁrst discovered by Hermite in  in the Journal Mathesis (see []). Inequality (.) was nowhere mentioned in the mathematical literature until . Beckenbach, a leading expert on the theory of convex functions, wrote that inequality (.) was proven by Hadamard in  (see []). In  Mitrinovič found Hermite’s note in Mathesis. That is why, inequality (.) was known as the Hermite-Hadamard inequality.

A functionf : [a,b]⊂R→Ris said to be convex if wheneverx,y∈[a,b] andt∈[, ], the following inequality holds:

ftx+ ( –t)y≤tf(x) + ( –t)f(y).

We say thatf is concave if (–f) is convex. This deﬁnition has its origins in Jensen’s results from [] and has opened up the most extended, useful and multi-disciplinary domain of mathematics, namely, convex analysis. Convex curves and convex bodies have appeared in mathematical literature since antiquity and there are many important results related to them.

In [], Miheşan introduced the class of (α,m)-convex functions in the following way: The functionf : [,b]→Ris said to be (α,m)-convex, where (α,m)∈[, ]_{, if for every}

x,y∈[,b] andt∈[, ], we have

ftx+m( –t)y≤tαf(x) +m –tαf(y).

This class is usually denoted byKα

m(b).

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In [], the concept of geometrically convex functions was introduced as follows.

Deﬁnition  A functionf :I⊂R+→R+is said to be a geometrically convex function if

fxty–t≤f(x)tf(y)–t for allx,y∈Iandt∈[, ].

In [], the deﬁnition ofm- and (α,m)-geometric convexity was introduced as follows.

Deﬁnition  Letf(x) be a positive function on [,b] andm∈(, ]. If

fxtym(–t)≤f(x)tf(y)m(–t)

holds for allx,y∈[,b] andt∈[, ], then we say that the functionf(x) ism-geometrically convex on [,b].

It is clear that when m= ,m-geometrically convex functions become geometrically convex functions.

Deﬁnition  Letf(x) be a positive function on [,b] and (α,m)∈(, ]×(, ]. If

fxtym(–t)≤f(x)tαf(y)m(–tα)

holds for all x,y∈ [,b] and t∈ [, ], then we say that the function f(x) is (α,m )-geometrically convex on [,b].

Ifα=m= , the (α,m)-geometrically convex function becomes a geometrically convex function on [,b].

Lemma  For x,y∈[,∞)and m,t∈(, ],if x<y and y≥,then

xtym(–t)≤tx+ ( –t)y. (.)

For some recent results connected with geometrically convex functions, see [–].

Deﬁnition  Leta,b∈R,a,b=  and|a| =|b|. Logarithmic mean for real numbers was introduced as follows:

L(a,b) = a–b ln|a|–ln|b|.

Theorem (see []) Let f,g: [a,b]→Rbe integrable functions,both increasing or both decreasing.Furthermore,let p: [a,b]→R+be an integrable function.Then

b

a

p(x)f(x)dx b

a

p(x)g(x)dx≤ b

a

p(x)dx b

a

p(x)f(x)g(x)dx. (.)

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and so are the following special cases of(.):



b–a b

a

f(x)dx b

a

g(x)dx≤ b

a

f(x)g(x)dx

and 



f(x)dx 



g(x)dx≤ 



f(x)g(x)dx.

In order to prove our main results, we need the following lemma (see []).

Lemma  Let f :I⊂R→Rbe a twice diﬀerentiable function on I◦,a,b∈I with a<b and f∈L[a,b].Then the following equality holds:

f(a) +f(b)

 –



b–a b

a

f(x)dx=(b–a)



 



t( –t)fta+ ( –t)bdt.

In [], some inequalities of Hermite-Hadamard type for diﬀerentiable convex mappings were proven using the following lemma.

Lemma  Let f :I⊂R→Rbe a twice diﬀerentiable function on I◦,a,b∈I◦with a<b.If f∈L[a,b],then



b–a b

a

f(x)dx–f

a+b



=(b–a)



 



m(t)fta+ ( –t)b+f( –t)a+tbdt,

where

m(t) = t

_, _t_∈_[, ),

( –t)_, _t_∈_[ , ].

In this paper, we establish some integral inequalities of Hermite-Hadamard type related to geometrically convex functions and (α,m)-geometrically convex functions.

2 Results for geometrically convex functions

We will establish some new results connected with the right-hand side of (.).

Theorem  Let f :I⊂R+→R+be a twice diﬀerentiable function on I◦,a,b∈I with a<b

and f∈L[a,b].If|f|qis geometrically convex and monotonically decreasing on[a,b]for

p> and t∈[, ],then we have

f(a) +_f(b)– 

b–a b

a

f(x)dx

≤(b–a)



––p√_π_{( +}_p₎ (_+p)



p

Lf(a)q,f(b)qq_, _(.)

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Proof From Lemma  with the properties of modulus and using the Hölder inequality, we have

f(a) +_f(b)– 

b–a b

a

f(x)dx

≤(b–a)

 



t( –t)fta+ ( –t)bdt

≤(b–a)







t–tpdt 

p 



fta+ ( –t)bqdt 

q

=(b–a)



––p√_π_{( +}_p₎ (

+p)



p 



fta+ ( –t)bqdt 

q

. (.)

We used the beta and gamma functions to evaluate the integral

 

t–tpdt= 



tp( –t)p_dt₌_β₍_p_{+ ,}_p_{+ ),}

β(x,x) = –x_β

 ,x

and β(x,y) =(x)(y) (x+y).

Thus, we have

β(p+ ,p+ ) = –(p+)(



)(p+ )

(_+p) , (.)

where(_) =√π.

Since|f|q_{is geometrically convex and monotonically decreasing on [}_a_,_b_{], we obtain}

atb–t≤ta+ ( –t)b,

fta+ ( –t)bq≤fatb–tq.

Therefore, we have

I = 



fta+ ( –t)bqdt

≤

 

fatb–tqdt

≤

 

f(a)tf(b)–tqdt

=Lf(a)q,f(b)q. (.)

By making use of inequalities (.) and (.) in (.), we obtain (.). This completes the

proof.

Theorem  Let f :I⊂R+→R+be a twice diﬀerentiable function on I◦,a,b∈I with a<b

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q≥and t∈[, ],then the following inequality holds:

f(a) +_f(b)– 

b–a b

a

f(x)dx≤(b–a)



 



p

f(b)·N(t,q),

where

N(t,q) =





t( –t) _|

f(a)|

|f(b)| qt

dt 

q .

Proof From Lemma  and using the well-known power-mean inequality, we have

f(a) +f(b)

 –



b–a b

a

f(x)dx

≤(b–a)

 



t( –t)fta+ ( –t)bdt

≤(b–a)







t( –t)dt

–_q  

t( –t)fta+ ( –t)bqdt 

q

=(b–a)



 

–q 



t( –t)fta+ ( –t)bqdt 

q .

Since|f|qis geometrically convex and monotonically decreasing on [a,b], we have

f(a) +_f(b)– 

b–a b

a

f(x)dx

≤(b–a)



 

–_q  

t( –t)f(a)tf(b)–tqdt 

q

=(b–a)



 

–_q

f(b)





t( –t) _|

f(a)|

|f(b)| qt

dt 

q ,

which completes the proof.

Corollary  In Theorem,since(_)p_{< ,}_{for p}_{> ,}_{we have}

f(a) +_f(b)– 

b–a b

a

f(x)dx≤(b–a)



 f

₍_b₎_N₍_t_,_q_).

Now, we will establish some new results connected with the left-hand side of (.).

Theorem  Let f :I⊂R+→R+be a twice diﬀerentiable function on I◦,a,b∈I with a<b

and f∈L[a,b].If|f|qis geometrically convex and monotonically decreasing on [a,b]

and t∈[, ],then we have the following inequality:

_b_–_a b a

f(x)dx–f

a+b



≤ (b–a)

(p+ )p

Lf(a)q,f(b)qq_,

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Proof From Lemma  and using the well-known Hölder integral inequality, we get

_b_–_a_abf(x)dx–f

a+b



≤(b–a)







m(t)pdt 

p 



fta+ ( –t)bqdt 

q

+





f( –t)a+tbqdt 

q

.

Since|f|q_{is geometrically convex and monotonically decreasing on [}_a_,_b_{], we know that} fort∈[, ],

fta+ ( –t)bq≤fatb–tq

≤f(a)tf(b)–tq.

Hence

_b_–_a b a

f(x)dx–f

a+b



≤ (b–a)

(p+ )p





f(a)tf(b)–tqdt 

q +





f(a)–tf(b)tqdt 

q

= (b–a)



(p+ )p

Lf(a)q,f(b)q



q_,

where we have used the fact that 



m(t)pdt= 





tpdt+ 

 

( –t)pdt=  p_(_p_{+ )},

Corollary  In Theorem,since 

(p+)



p < ,for p> ,we have

_b_–_a b a

f(x)dx–f

a+b



≤(b–a)



Lf(a)q,f(b)q



q_.

Theorem  Let f :I◦⊂R+→R+ be a twice diﬀerentiable function on I◦,a,b∈I with

a<b and f∈L[a,b].If|f|q is geometrically convex and monotonically decreasing on

[a,b]and q≥,then we have

_b_–_a b a

f(x)dx–f

a+b



≤(b–a)



 



p

f(a)·A(t,q) +f(b)·B(t,q),

where

A(t,q) =





m(t)· _|

f(b)|

|f(a)| qt

dt 

q

, B(t,q) =





m(t)· _|

f(a)|

|f(b)| qt

dt 

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Proof From Lemma  and using the well-known power-mean inequality, we get

_b_–_a b a

f(x)dx–f

a+b



≤(b–a)







m(t)dt 

p 



m(t)fta+ ( –t)bqdt 

q

+





m(t)f( –t)a+tbqdt 

q

.

Since|f|q_{is geometrically convex and monotonically decreasing on [}_a_,_b_{], we have}

a+b



≤(b–a)

()p





m(t)f(a)tf(b)–tqdt 

q

+





m(t)f(a)–tf(b)tqdt 

q

=(b–a)



()p





m(t)·

|f(b)|

|f(a)| qt

dt 

q

+





m(t)·

|f(a)|

|f(b)| qt

dt 

q

,

where we have used the fact that 



m(t)dt= 





tdt+ 

 

( –t)dt=  ,

Corollary  In Theorem,since(_)p_{< ,}_{for p}_{> ,}_{we have the following inequality}_:

a+b



≤(b–a)

 f

₍_a₎_·_A₍_t_,_q_{) +}_f₍_b₎_·_B₍_t_,_q₎_.

3 Results for(

α

,m)-geometrically convex functions

Theorem  Let f : [,b]→R+ be a twice diﬀerentiable function, a∈[,b]and f∈

L[a,b].If|f|q is(α,m)-geometrically convex and monotonically decreasing on[a,b]for

b≥,p> , (α,m)∈(, ]and t∈(, ],we have

f(a) +_f(b)– 

b–a b

a

f(x)dx≤(b–a)



––p√_π_{( +}_p₎ (

+p)



p

f(b)mM(t,q,α),

where_p+_q= and

M(t,q,α) =





_| f(a)|

|f(b)|m qtα

dt 

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Proof From Lemma  with the properties of modulus and using the Hölder inequality, we have

f(a) +_f(b)– 

b–a b

a

f(x)dx

≤(b–a)

 



t( –t)fta+ ( –t)bdt

≤(b–a)







t–tpdt 

p 



fta+ ( –t)bqdt 

q

=(b–a)



––p√_π_{( +}_p₎ (

+p)



p 



fta+ ( –t)bqdt 

q .

Since|f|q _{is (}_α_,_m_{)-geometrically convex and monotonically decreasing on [}_a_,_b_{], by} using (.) we obtain

I = 



fta+ ( –t)bqdt

≤

 

fatbm(–t)qdt

≤

 

f(a)tαf(b)m(–tα)qdt

=f(b)mq 



_| f(a)|

|f(b)|m qtα

dt.

So, we have

f(a) +_f(b)– 

b–a b

a

f(x)dx

≤(b–a)



––p√_π_{( +}_p₎ (_+p)



p

_f₍_b₎mq 



|f(a)|

|f(b)|m qtα

dt 

q

=(b–a)

_|_f₍_b₎_|m 

––p√_π_{( +}_p₎ (_+p)



p 



|f(a)|

|f(b)|m qtα

dt 

q ,

Corollary  In Theorem, (i)If|f(a)|<|f(b)|m_,_{then we get}

f(a) +_f(b)– 

b–a b

a

f(x)dx

≤(b–a)|f(b)|m



––p√π ( +p) (_+p)



p 



_| f(a)|

|f(b)|m qtα

dt 

q

≤(b–a)|f(b)|m



––p√π ( +p) (_+p)



p 



_| f(a)|

|f(b)|m qαt

dt 

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=(b–a)

_|_f₍_b₎_|m 

––p√π ( +p) (_+p)



p

×

_|

f(a)|qα_–_|_f₍_b₎_|mqα

|f(b)|mqα_[_ln_|_f₍_a₎_|qα_–_ln_|_f₍_b₎_|mqα_]



q

=(b–a)

_|_f₍_b₎_|m(–α)



––p√_π_{( +}_p₎ (

+p)



p

Lf(a)qα,f(b)mqαq_,

where we used the fact that if <μ≤,  <t andα≤,we have

μtα≤μαt.

(ii)If|f(a)|=|f(b)|m_,_{then we get}

f(a) +_f(b)– 

b–a b

a

f(x)dx

≤(b–a)|f(b)|m



––p√π ( +p) (_+p)



p .

Theorem  Let f : [,b]→R+ be a twice diﬀerentiable function, a∈[,b]and f∈

L[a,b].If|f|and|f|qare(α,m)-geometrically convex and monotonically decreasing on

[a,b]for a,b≥,p> , (α,m)∈(, ]and t∈[, ],we have

a+b



≤(b–a)



 

f(b)m 





K dt+f(a)m 





G dt

+ 

(p+ )p

f(b)m



 

Kqdt 

q

+f(a)m



 

Gqdt 

q

, (.)

where K= (_|f|f₍_b(a₎_|)m| )t

α

,G= (_|f|f₍(_ab₎_|)m| )t

α and

p+



q= .

Proof By using Lemma  and the properties of absolute value, we have

_b_–_a b a

f(x)dx–f

a+b



≤(b–a)









tfta+ ( –t)bdt+ 





tf( –t)a+tbdt

+ 

 

( –t)fta+ ( –t)bdt+ 

 

( –t)f( –t)a+tbdt

=(b–a)



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Sincet_{is increasing on [,}

] and|f|is decreasing, by applying the Chebyshev inequality

toIandIand by applying the Hölder inequality toIandI, we get

a+b



≤(b–a)

    

tdt 





fta+ ( –t)bdt

+  





tdt 





f( –t)a+tbdt

+



 

( –t)pdt 

p 

 

fta+ ( –t)bqdt  q +   

( –t)p_dt 

p 

 

_f_{( –}_t₎_a₊_tbq dt



q

=(b–a)

      

fta+ ( –t)bdt+ 





f( –t)a+tbdt

+ 

(p+ )p_

p+

p



 

fta+ ( –t)bqdt  q +   

q

.

As|f|is decreasing,|f|and|f|q_{are (}_α_,_m_{)-geometrically convex, from (.) we have}

_b_–_a b a

f(x)dx–f

a+b



≤(b–a)

     

fatbm(–t)dt+ 





fam(–t)btdt

+ 

(p+ )p_

p+

p



 

fatbm(–t)qdt  q +   

fam(–t)btqdt 

q

≤(b–a)

     

_f₍_a₎tα

f(b)m(–tα)dt+ 





_f₍_a₎m(–tα₎

f(b)tαdt

+ 

(p+ )p_

p+

p



 

f(a)tαf(b)m(–tα)qdt  q +   

_f₍_a₎m(–tα₎

f(b)tαqdt 

q

,

(11)

Corollary  Under the conditions of Theorem,if we choose fis symmetric about a+_b,

then we have

a+b



≤(b–a)|f(b)|m

     

K dt+ 

(p+ )p



 

Kqdt 

q

.

Proof From (.) and by using the symmetric property off, we get

_b_–_a b a

f(x)dx–f

a+b



≤(b–a)

     

fta+ ( –t)bdt+ 





f( –t)a+tbdt

+ 

(p+ )p_pp+



 

fta+ ( –t)bqdt  q +   

q

=(b–a)

      

fta+ ( –t)bdt

+ 

(p+ )p_

p+

p



 

fta+ ( –t)bqdt 

q

=(b–a)

      

_f_ta_{+ ( –}_t₎_b_dt

+ 

(p+ )p_p



 

fta+ ( –t)bqdt 

q

.

By using (α,m)-geometric convexity of|f|and|f|qand similar calculations, we get

a+b



≤(b–a)

     

fat_bm(–t)_dt₊ 

(p+ )p_p



 

fat_bm(–t)q_dt



q

≤(b–a)

     

_f₍_a₎tα

f(b)m(–tα)dt

+ 

(p+ )p



 

f(a)tαf(b)m(–tα)qdt 

q

=(b–a)

_|_f₍_b₎_|m      

|f(a)|

|f(b)|m tα

dt+ 

(p+ )p



 

|f(a)|

|f(b)|m qtα

dt 

q

.

(12)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ÇY and MG carried out the design of the study and performed the analysis. MEÖ participated in its design and coordination. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, K. K. Education Faculty, Atatürk University, Campus, Erzurum, 25240, Turkey.}2_Graduate School of Natural and Applied Sciences, A ˘gri ˙Ibrahim Çeçen University, A ˘gri, Turkey.

Received: 20 September 2013 Accepted: 11 April 2014 Published:12 May 2014

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10.1186/1029-242X-2014-180