convex functions - Journal of Inequalities and Applications

Xi et al. Journal of Inequalities and Applications 2014, 2014:100 http://www.journalofinequalitiesandapplications.com/content/2014/1/100

RESEARCH

Open Access

Some inequalities for (h, m)-convex functions Bo-Yan Xi1* , Shu-Hong Wang1 and Feng Qi1,2 * Correspondence: [email protected]; [email protected] 1 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia Autonomous Region 028043, China Full 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 definition is well known in the literature. Definition  A function f : I ⊆ R = (–∞, ∞) → R is said to be convex if f tx + ( – t)y ≤ 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 x + x + x   x + x x  + x x + x   f +f +f . ≥    

f (x ) + f (x ) + f (x ) + f

()

Theorem  ([, Popoviciu inequality]) If f is a convex function on I and x , x , . . . , xn ∈ I with n ≥ , then n i=

n f f (xi ) + n–

n  xi + x j  . xi ≥ f n i= n –  i  with i = , , . . . , n and n ≥ , we have f

n n wi  i– m wi xi ≤ mi– h f (xi ), Wn i= Wn i=

()

where Wn = ni= wi . If h is sub-multiplicative and f ∈ SMV((h, m), [, b]), then the inequality () is reversed. Proof Assume that wi = Wwin for i = , , . . . , n. When n = , taking t = w and  – t = w in Definition  gives the inequality () clearly. Suppose that the inequality () holds for n = k, i.e., f

k

mi– wi xi ≤

i=

k

mi– h wi f (xi ).

When n = k + , letting k = f

k+

()

i=

m

i–

wi xi

i=

k+

i=

wi and making use of () result in

wi =f + mk m xi k i=

k+ i– wi ≤ h w f (x ) + mh(k )f m xi k i= w x

k+

i–

k+ wi i– ≤ h w f (x ) + mh(k ) m h f (xi ). k i= Since h is a super-multiplicative function, it follows that h(k )h

wi k

≤ h wi

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


Page 5 of 12

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

n

mi– wi xi ≤

i=

n

mi– h(wi )f (xi );

()

i=

if w = w = · · · = wn , we have

.

f

n n   i– m wi xi ≤ h mi– f (xi ); n i= n i=

()

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((t s , m), [, b]) is valid if and only if for all xi ∈ [, b] and wi >  with i = , , . . . , n and n ≥  f

n n wi s  i– i– m xi ≤ m f (xi ), Wn i= Wn i=

where Wn =

()

n

i= wi .

Corollary  Under the conditions of Corollary , if h(t) = t s for s ∈ (, ], then f

n n  i–  i– m xi ≤ s m f (xi ). n i= n i=

()

If f ∈ SMV((h, m), [, b]), then the inequality () is reversed. Theorem  Let h : [, ] → R be a super-multiplicative function, m ∈ (, ], and n ≥ . b If f ∈ SMX((h, m), [, mn– ]), then for all xi ∈ [, b] and wi >  with i = , , . . . , n, f

n n wi xi  i– wi xi ≤ m h f , Wn i= Wn mi– i=

()

where Wn = ni= wi . b If h is sub-multiplicative and f ∈ SMV((h, m), [, mn– ]), then the inequality () is reversed. Proof Putting yi = f

xi mi–

for i = , , . . . , n, then from inequality (), we have

n n  i–  wi xi = f m wi yi Wn i= Wn i= ≤

n i=

n wi wi xi i– m h m h f (yi ) = f . Wn Wn mi– i= i–

The proof of Theorem  is complete.


Page 6 of 12

b Corollary  For m ∈ (, ], s ∈ (, ], and n ≥ , the assertion f ∈ SMX((t s , m), [, mn– ]) is valid if and only if for all xi ∈ [, b] and wi >  with i = , , . . . , n the inequality

f

n n xi wi s  i– wi xi ≤ m f Wn i= Wn mi– i=

is valid, where Wn =

()

n

i= wi .

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

n

wi xi ≤

i=

.

mi– h(wi )f

i=

xi ; mi–

()

if w = w = · · · = wn , then f

.

n

n n  xi  i– xi ≤ h m f ; n i= n i= mi–

()

b if h is sub-multiplicative and f ∈ SMV((h, m), [, mn– ]), then the inequalities () and () are reversed.

Corollary  Under the conditions of Corollary , . if h(t) = t s for s ∈ (, ], then f

.

n n  i– xi  xi ≤ s m f ; n i= n i= mi–

()

b if f ∈ SMV((h, m), [, mn– ]), 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

f (xi ) –

i=

n– j=

– m

j

n i=

f

n+i–  j–i m xj n j=i

k– – n k+i–   – h(/n) j j–i m f m xj , ≥ h(/k) k j=i j= i=

()

where xn+ = x , . . . , xn– = 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

n k+i– n k+i– k–    j–i j–i j m xj ≤ h m f (xj ) = h m f (xi ) k j=i k i= j=i k j= i=

()


Page 7 of 12

and n

f

i=

n+i– n n+i–   j–i m xj ≤ h mj–i f (xj ) n j=i n i= j=i

n n–  j =h m f (xi ). n j= i=

()

If h( n ) = , then, from the inequality (), the inequality () holds. If h( n ) ≤ , it is easy to see that n i=

f

k+i–  j–i m xj k j=i

n k–  j m f (xi ) ≤h k j= i= k– n

n h(/k)  j = m f (xi ) – h f (xi )  – h(/n) j= n i= i=

k– n n– – n n+i–  h(/k) j j j–i m f (xi ) – m f m xj . ≤  – h(/n) j= n j=i i= j= i=

The proof of Theorem  is complete. Corollary  Under the conditions of Theorem , let x¯ n = . When m = , we have n

f (xi ) – f

i=

.

f (xi ) – f

i=

i= xi .

k+i–

n n  – h(/n)   xi ≥ f xj . n i= kh(/k) i= k j=i

()

n n  – h(/n) xi + xi+  . xi ≥ f n i= h(/) i= 

()

When m =  and k = n – , we have n i=

.

n

When m =  and k = , we have n

.

 n

f (xi ) – f

n n  – h(/n) n¯xn – xi  . xi ≥ f n i= (n – )h(/(n – )) i= 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 = i for i = , , . . . , n. , , . . . , n, a = n ni= ai , and bi = na–a n–


Page 8 of 12

Corollary  Under the conditions of Theorem , . if h(t) = t s for s ∈ (, ], then n

f (xi ) –

n–

i=

– m

j

n

j=

f

i=


k– – n k+i–

 k s (ns – ) j j–i m f m xj ; ≥ ns k j=i j= i= .

if h(t) = t s for s ∈ (, ] and m = , then n

f (xi ) – f

i=

.

k+i–

n n k s– (ns – )   xi ≥ f xj ; n i= ns k j=i i=

()

if h(t) = t and m = , then n

f (xi ) – f

i=

.

()

k+i–

n n n–   xi ≥ f xj ; n i= n i= k j=i

()

if f ∈ SMV((h, m), [, b]), then the inequalities () to () are reversed.

Theorem  Let h : [, ] → [, ] be a super-multiplicative function and let m ∈ (, ] and b n ≥ . If f ∈ SMX((h, m), [, mn– ]), then for all xi ∈ [, b] with i = , , . . . , n and  ≤ k ≤ n and for  , . . . , k ∈ N, we have n

f (xi ) –

i=

n–

– m

j

j=

n

f

i=


k– –  – h(/n) j m ≥ n– h(/k) j= k– ≤