Some inequalities for [InlineEquation not available: see fulltext.]-convex functions

Abstract

In the paper, the authors give some inequalities of Jensen type and Popoviciu type for [InlineEquation not available: see fulltext.]-convex functions and apply these inequalities to special means.

MSC: 26A51, 26D15, 26E60.

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R E S E A R C H Open Access

Some inequalities for (h, m)-convex functions

Bo-Yan Xi1*, Shu-Hong Wang1 and Feng Qi1,2

*Correspondence: mailto:[email protected]

Web End [email protected] ; mailto:[email protected]

Web End [email protected]

1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia Autonomous Region 028043, ChinaFull list of author information is available at the end of the article

Abstract

In the paper, the authors give some inequalities of Jensen type and Popoviciu type for (h, m)-convex functions and apply these inequalities to special means.

MSC: Primary 26A51; secondary 26D15; 26E60

Keywords: convex function; (h, m)-convex function; Jensen inequality; Popoviciu inequality

1 Introduction

The following denition is well known in the literature.

Denition A function f : I

R = (, )

R is said to be convex if

f

tx + ( t)y[parenrightbig] tf (x) + ( t)f (y) ()

holds for all x, y I and t [, ].

We cite the following inequalities for convex functions.

Theorem ([, p.]) If f is a convex function on I and x, x, x I, then

f (x) + f (x) + f (x) + f

x + x + x

[parenrightbigg]

x + x

[parenrightbigg][bracketrightbigg]. ()

Theorem ([, Popoviciu inequality]) If f is a convex function on I and x, x, . . . , xn I with n , then

n

i=f (xi) + nn f [parenleftBigg] n

n

i=xi

f

x + x

[parenrightbigg]+ f

x + x

[parenrightbigg]+ f

[parenrightBigg]

n

. ()

Theorem ([, Generalized Popoviciu inequality]) If f is a convex function on I and a, a, . . . , an I for n , then

(n )

n

i<jf

xi + xj

i=f (bi) n(n )f (a) +

n

i=f (ai), ()

where a = n

ni= ai and bi = naain for i = , , . . . , n.

2014 Xi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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The above inequalities were generalized as follows.

Theorem ([]) If f is a convex function on I and x, x, . . . , xn I for n , then

n

i=f (xi) f

[parenleftBigg]

n

n

k=xk

[parenrightBigg]

n n

n

i=f

xi + xi+

[parenrightbigg]

()

and

(n )

n

i=f (bi) n

[bracketleftBigg]

n

i=f (ai) f (a)

, ()

ni= ai, and bi = naain for i = , , . . . , n.

Denition ([]) Let s (, ]. A function f :

R = [, )

where xn+ = x, a = n

R is said to be s-convex in

the second sense if

f

x + ( )y[parenrightbig] sf (x) + ( )sf (y) ()

holds for all x, y I and [, ].

The following inequalities for s-convex functions were established.

Theorem ([, Theorem .]) If f is nonnegative and s-convex in the second sense on I and if x, x, . . . , xn I for n , then

n

i=f (xi) f

[parenleftBigg]

n

n

i=xi

[parenrightBigg]

s(ns )

n

n

i=f

xi + xi+

, ()

where x = xn+.

Theorem ([, Theorem .]) If f is nonnegative and s-convex in the second sense on I and a, a, . . . , an I for n , then

ns

[parenrightbig]

n

i=bi ns[bracketleftBigg]

n

i=f (ai) f (a)

, ()

ni= ai and bi = naain for i = , , . . . , n.

The concept of h-convex functions below was innovated as follows.

Denition ([, Denition ]) Let I, J

R be intervals, (, ) J, and h : J

R be a

where a = n

nonnegative function. A function f : I

R is called h-convex, or as we say, f belongs to the class SX(h, I), if f is nonnegative and

f

tx + ( t)y[parenrightbig] h(t)f (x) + h( t)f (y) ()

for all x, y I and t [, ].

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Denition ([, Section ]) A function h : J

R is said to be a super-multiplicative on

an interval J if

h(xy) h(x)h(y) ()

is valid for all x, y J. If the inequality () reverses, then f is said to be a sub-multiplicative function on J.

The following inequalities were established for f SX(h, I).

Theorem ([, Theorem ]) Let w, . . . , wn for n be positive real numbers. If h is a nonnegative and super-multiplicative function and if f SX(h, I) and x, . . . , xn I, then

f

[parenleftBigg]

Wn

n

i=wixi

[parenrightBigg]

n

i=h

[parenleftbigg]

wi Wn

f (xi), ()

where Wn =

ni= wi. If h is sub-multiplicative and f SV(h, I), then the inequality () is reversed.

Theorem ([, Theorem ]) Let h be a nonnegative and super-multiplicative function. If f SX(h, I) and x, . . . , xn I, then

n

i=f (xi) f

[parenleftBigg]

n

n

i=xi

[parenrightBigg]

h(/n) h(/)

n

i=f

xi + xi+

, ()

where xn+ = x. The inequality () is reversed if f SV(h, I).

Theorem ([, Theorem ]) Let h be a nonnegative and super-multiplicative function. If f SX(h, I) and x, . . . , xn I, then

h[parenleftbigg] n

[parenrightbigg][bracketrightbigg]

n

i=f (bi) (n )h

[parenleftbigg]

n

[parenrightbigg][bracketleftBigg]

n

i=f (ai) f (a)

, ()

ni= ai and bi = naain for i = , , . . . , n and n . The inequality () is reversed if f SV(h, I).

Two new kinds of convex functions were introduced as follows.

Denition ([]) For f : [, b]

R and m (, ], if

f

tx + m( t)y[parenrightbig] tf (x) + m( t)f (y) ()

is valid for all x, y [, b] and t [, ], then we say that f (x) is an m-convex function on [, b].

Denition ([]) Let J

R be an interval, (, ) J, h : J

R be a nonnegative func-

where a = n

tion. We say that f : [, b]

R is an (h, m)-convex function, or say, f belongs to the class

SMX((h, m), [, b]), if f is nonnegative and, for all x, y [, b] and t [, ] and for some

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m (, ], we have

f

tx + m( t)y[parenrightbig] h(t)f (x) + mh( t)f (y). ()

If the inequality () is reversed, then f is said to be (h, m)-concave and denoted by f

SMV((h, m), [, b]).

Recently the h- and (h, m)-convex functions were generalized and some properties and inequalities for them were obtained in [, ].

The aim of this paper is to nd some inequalities of Jensen type and Popoviciu type for (h, m)-convex functions.

2 Inequalities of Jensen type and Popoviciu type

Now we are in a position to establish some inequalities of Jensen type and Popoviciu type for (h, m)-convex functions.

Theorem Let h : [, ]

R be a super-multiplicative function and m (, ]. If f

SMX((h, m), [, b]), then for all xi [, b] and wi > with i = , , . . . , n and n , we have

f

[parenleftBigg]

Wn

i= miwixi

[parenrightBigg]

[parenleftbigg]

wi Wn

n

n

i= mih

f (xi), ()

ni= wi.

If h is sub-multiplicative and f SMV((h, m), [, b]), then the inequality () is reversed.

Proof Assume that w i = wiWn for i = , , . . . , n.When n = , taking t = w and t = w in Denition gives the inequality () clearly. Suppose that the inequality () holds for n = k, i.e.,

f

[parenleftBigg]

k

where Wn =

k

i= miw ixi

[parenrightBigg]

i= mih

w i

f (xi). ()

When n = k + , letting k =

k+i= w i and making use of () result in

k+

f

[parenleftBigg]

i= miw ixi

[parenrightBigg] = f [parenleftBigg]w x + m

k

k+

i=mi w i k xi[parenrightBigg]

h

w

f (x) + mh( k)f

[parenleftBigg]

k+

i=mi w i k xi[parenrightBigg]

h

w

f (x) + mh( k)

k+

i= mih

[parenleftbigg]

w i k

f (xi).

Since h is a super-multiplicative function, it follows that

h( k)h

[parenleftbigg]

w i k

[parenrightbigg]

h

w i

[parenrightbig]

for i = , , . . . , n. Namely, when n = k + , the inequality () holds. By induction, Theorem is proved.

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Corollary Under the conditions of Theorem ,. if Wn = , we have

f

[parenleftBigg]

n

n

i= miwixi

[parenrightBigg]

i=mih(wi)f (xi); ()

. if w = w = = wn, we have

f

[parenleftBigg]

n

n

[parenrightbigg]

n

n

i= miwixi

[parenrightBigg] h[parenleftbigg]

i=mif (xi); ()

. if h is sub-multiplicative and f SMV((h, m), [, b]), then the inequalities () and() are reversed.

Corollary For m (, ] and s (, ], the assertion f SMX((ts, m), [, b]) is valid if and only if for all xi [, b] and wi > with i = , , . . . , n and n

f

[parenleftBigg]

Wn

n

n

i= mixi

[parenrightBigg]

i=mi

[parenleftbigg]

wi Wn

sf (xi), ()

ni= wi.

Corollary Under the conditions of Corollary , if h(t) = ts for s (, ], then

f

[parenleftBigg]

n

where Wn =

ns

n

n

i= mixi

[parenrightBigg]

i=mif (xi). ()

If f SMV((h, m), [, b]), then the inequality () is reversed.

Theorem Let h : [, ]

R be a super-multiplicative function, m (, ], and n .

If f SMX((h, m), [,

bmn ]), then for all xi [, b] and wi > with i = , , . . . , n,

f

[parenleftBigg]

Wn

n

i=wixi

[parenrightBigg]

i= mih

[parenleftbigg]

wi Wn

f[parenleftbigg]xi mi

, ()

n

ni= wi.

If h is sub-multiplicative and f SMV((h, m), [,bmn ]), then the inequality () is re-

versed.

Proof Putting yi = xi

mi for i = , , . . . , n, then from inequality (), we have

where Wn =

f

[parenleftBigg]

Wn

n

i=wixi

[parenrightBigg] = f [parenleftBigg]

Wn

n

i= miwiyi

[parenrightBigg]

n

i= mih

[parenleftbigg]

wi Wn

f (yi) =n

i= mih

[parenleftbigg]

wi Wn

f[parenleftbigg]xi mi

.

The proof of Theorem is complete.

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Corollary For m (, ], s (, ], and n , the assertion f SMX((ts, m), [,

bmn ]) is

valid if and only if for all xi [, b] and wi > with i = , , . . . , n the inequality

f

[parenleftBigg]

Wn

n

n

i=wixi

[parenrightBigg]

i=mi

[parenleftbigg]

wi Wn

s f

[parenleftbigg]

xi mi

()

is valid, where Wn =

ni= wi.

Corollary Under the conditions of Theorem ,. if Wn = , then

f

[parenleftBigg]

n

n

i=wixi

[parenrightBigg]

i= mih(wi)f

[parenleftbigg]

xi mi

; ()

. if w = w = = wn, then

f

[parenleftBigg]

n

n

n

i=xi

[parenrightBigg] h[parenleftbigg]

n

i= mif

[parenleftbigg]

xi mi

; ()

. if h is sub-multiplicative and f SMV((h, m), [,

bmn ]), then the inequalities ()

and () are reversed.

Corollary Under the conditions of Corollary ,. if h(t) = ts for s (, ], then

f

[parenleftBigg]

n

n

n

i=xi

[parenrightBigg]

ns

i= mif

[parenleftbigg]

xi mi

; ()

bmn ]), then the inequality () is reversed.

Theorem Let h : [, ] [, ] be a super-multiplicative function and let m (, ] and n . If f SMX((h, m), [, b]), then for all xi [, b] with i = , , . . . , n and k n, we have

n

. if f SMV((h, m), [,

i=f (xi)

n

j=mj

n

i=f

n

n+i

j=imjixj

k

j=mj

n

i=f

h(/n) h(/k)

k k+i

j=imjixj

, ()

where xn+ = x, ..., xn = xn.

If h is sub-multiplicative and f SMV((h, m), [, b]), then the inequality () is reversed.

Proof By using the inequality (), we have

n

i=f

k k+i

j=imjixj

[parenrightBigg] h[parenleftbigg]

k

[parenrightbigg]

n

i=

k

[parenrightbigg][parenleftBigg]

k

j=mj

[parenrightBigg]

k+i

j=imjif (xj) = h

n

i=f (xi) ()

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and

n

i=f

[parenleftBigg]

n

[parenrightBigg] h[parenleftbigg]

n

[parenrightbigg]

n

i=

n+i

j=imjixj

n+i

j=imjif (xj)

= h

[parenleftbigg]

n

[parenrightbigg][parenleftBigg]

n

j=mj

[parenrightBigg]

n

i=f (xi). ()

If h( n) = , then, from the inequality (), the inequality () holds. If h( n) , it is easy to see that

n

i=f

k k+i

j=imjixj

[parenrightBigg]

n

h

k

[parenrightbigg][parenleftBigg]

k

j=mj

[parenrightBigg]

i=f (xi)

= h(/k)

h(/n)

k

j=mj

[parenrightBigg][bracketleftBigg]

n

n

i=f (xi) h

[parenleftbigg]

n

[parenrightbigg]

i=f (xi)

[bracketrightBigg]

h(/k) h(/n)

[parenrightBigg][bracketleftBigg]

n

i=f (xi)

n

j=mj

n

i=f

n

n+i

j=imjixj

[parenrightBigg][bracketrightBigg].

k

j=mj

The proof of Theorem is complete.

Corollary Under the conditions of Theorem , let xn = n

ni= xi.

. When m = , we have

i=f (xi) f

n

n

i=xi

[parenrightBigg]

h(/n) kh(/k)

n

n

i=f

k k+i

j=ixj

. ()

. When m = and k = , we have

i=f (xi) f

[parenleftBigg]

n

n

i=xi

[parenrightBigg]

h(/n) h(/)

n

n

i=f

xi + xi+

. ()

. When m = and k = n , we have

n

n

i=xi

[parenrightBigg]

h(/n)(n )h(/(n ))

n

i=f (xi) f

[parenleftBigg]

n

i=f

nxn xi n

. ()

. If h is sub-multiplicative and f SMV((h, m), [, b]), then the inequalities () to

() are reversed.

Remark The inequality () can be deduced from applying () to ai = xi for i = , , . . . , n, a = n

ni= ai, and bi = naain for i = , , . . . , n.

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Corollary Under the conditions of Theorem ,. if h(t) = ts for s (, ], then

n

i=f (xi)

n

j=mj

n

i=f

n

n+i

j=imjixj

k

j=mj

n

i=f

ks(ns )

ns

k k+i

j=imjixj

; ()

. if h(t) = ts for s (, ] and m = , then

n

i=f (xi) f

n

n

i=xi

[parenrightBigg]

ks(ns )

ns

n

i=f

k k+i

j=ixj

; ()

. if h(t) = t and m = , then

i=f (xi) f

n

n

i=xi

[parenrightBigg]

n n

n

i=f

k k+i

j=ixj

n

; ()

. if f SMV((h, m), [, b]), then the inequalities () to () are reversed.

Theorem Let h : [, ] [, ] be a super-multiplicative function and let m (, ] and n . If f SMX((h, m), [,

bmn ]), then for all xi [, b] with i = , , . . . , n and k n and for , . . . , k

N, we have

n

i=f (xi)

n

j=mj

n

i=f

n

n+i

j=imjixj

h(/n)

n k

h(/k)

k

j=mj

k

i=f

k k+i

j=imjix j

, ()

<< kn

where k+ = , ..., k = k.

If h is sub-multiplicative and f SMV((h, m), [, b]), then the inequality () is reversed.

Proof By the inequality (), we have

<< kn

k

i=f

k k+i

j=imjix j

[parenrightBigg]

k

[parenrightbigg] [summationdisplay]

k+i

j=imjif (x j)

= h

h

<< kn

k

i=

k

[parenrightbigg][parenleftBigg]

k

j=mj

[parenrightBigg] [summationdisplay]

i=f (x j)

<< kn

k

n

=

n k

h

k

[parenrightbigg][parenleftBigg]

k

j=mj

[parenrightBigg]

i=f (xi). ()

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If h( n) = , then, from the inequality (), the inequality () holds. If h( n) , using () and (), we have

<< kn

k

i=f

k k+i

j=imjix j

[parenrightBigg]

n

n k

h

k

[parenrightbigg][parenleftBigg]

k

j=mj

[parenrightBigg]

i=f (xi)

k

j=mj

=

n k

h(/k) h(/n)

[parenrightBigg][bracketleftBigg]

n

i=f (xi) h

[parenleftbigg]

n

[parenrightbigg]

n

i=f (xi)

[bracketrightBigg]

n k

h(/k) h(/n)

[parenrightBigg][bracketleftBigg]

n

i=f (xi)

n

j=mj

n

i=f

n

n+i

j=imjixj

[parenrightBigg][bracketrightBigg].

k

j=mj

The proof of Theorem is complete.

Corollary Under the conditions of Theorem , let xn = n

ni= xi.

. When m = , we have

n

i=f (xi) f

n

n

i=xi

[parenrightBigg]

h(/n)

n k

h(/k)

kk

j=

x j

. ()

<< kn

f

. When m = and k = , we have

n

i=f (xi) f

[parenleftBigg]

n

n

i=xi

[parenrightBigg]

h(/n) (n )h(/)

i<jnf

xi + xj

. ()

. When m = and k = n , we have

n

n

i=xi

[parenrightBigg]

h(/n)(n )h(/(n ))

n

i=f (xi) f

[parenleftBigg]

n

i=

nxn xi n

. ()

. If h is sub-multiplicative and f SMV((h, m), [, b]), then the inequalities () to

() are reversed.

Corollary Under the conditions of Theorem ,

. if h(t) = ts for s (, ], then

n

i=f (xi)

n

j=mj

n

i=f

n

n+i

j=imjixj

ks(ns )

n k

ns

k

j=mj

k

i=f

k k+i

j=imjix j

; ()

<< kn

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. if m = and h(t) = ts for s (, ], we have

n

i=f (xi) f

n

n

i=xi

[parenrightBigg]

ks(ns )

n k

ns

kk

j=

x j

; ()

<< kn

f

. if m = and h(t) = t, then

n

i=f (xi) f

n

n

i=xi

[parenrightBigg]

k(n )

n k

n

kk

j=

x j

; ()

. if f SMV((h, m), [, b]), then the inequalities () to () are reversed.

3 Applications to means

In what follows we will apply the theorems and corollaries in the above section to establish inequalities for some special means.

For r

R, r = , and m, s (, ], let f (x) = xr for x

R+ and h(t) = ts for t [, ]. Then . if r and < m , or if r < and m = , we have

tx + m( t)y

r txr + ( t)(my)r tsxr + m( t)syr

for x, y

<< kn

f

R+;

. if < r , < m , and s = , we have

tx + m( t)y

r txr + ( t)(my)r txr + m( t)yr

for x, y

R+.

Using Denition yields the following:

. if r and < m , or if r < and m = , the function f (x) = xr SMX((ts, m),

R+);

. if < r , < m , and s = , the function f (x) = xr SMV((t, m),

R+).

By virtue of Corollary , we obtain the following results.

Theorem Let n and xi

R+ for i = , , . . . , n, let r

R with r = and m, s (, ],

and let , . . . , k

N for k n and k+ = , ..., k = k. . If r and < m , or if r < and m = , then we have

n

i=xri

n

j=mj

n

i=

n

n+i

j=imjixj

r

ks(ns )

n k

ns

k

j=mj

k

i=

k k+i

j=imjix j

r; ()

<< kn

. if r or r < and if m = , we have

n

i=xri

n

n

i=xi

rks(ns )

n k

ns

r; ()

<< kn

kk

j=

x j

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. if r or r < and if m = s = , then

n

i=xri

n

n

i=xi

rk(n )

n k

n

r; ()

<< kn

kk

j=

x j

. if < r , < m , and s = , then the inequality () are reversed.

Corollary Under the conditions of Theorem , when k+ = , ..., k = k, we have the following conclusions.

. If r = , we have

n

i=xi

n

j=mj

n

i=

n

n+i

j=imjixj

ks(ns )

n k

ns

k

j=mj

k

i=

k k+i

j=imjix j

; ()

<< kn

. if r = and m = , we have

n

i=xi

n

n

i=xi

ks(ns )

n k

ns

kk

j=

x j

; ()

<< kn

. if r = and m = s = , then

n

i=xi

n

n

i=xi

k(n )

n k

n

. ()

<< kn

kk

j=

x j

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the manuscript and read and approved the nal manuscript.

Author details

1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia Autonomous Region 028043, China. 2Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, 300387, China.

Acknowledgements

The authors appreciate anonymous referees for their valuable comments on and careful corrections to the original version of this paper. This work was partially supported by the NNSF under Grant No. 11361038 of China and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159, China.

Received: 11 October 2013 Accepted: 18 February 2014 Published: 4 March 2014

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doi:10.1186/1029-242X-2014-100Cite this article as: Xi et al.: Some inequalities for (h, m)-convex functions. Journal of Inequalities and Applications 2014 2014:100.