Some inequalities for -convex functions
© Xi et al.; licensee Springer. 2014
Received: 11 October 2013
Accepted: 18 February 2014
Published: 4 March 2014
In the paper, the authors give some inequalities of Jensen type and Popoviciu type for -convex functions and apply these inequalities to special means.
MSC: 26A51, 26D15, 26E60.
The following definition is well known in the literature.
holds for all and .
We cite the following inequalities for convex functions.
Theorem 1 ([, p.6])
Theorem 2 ([, Popoviciu inequality])
Theorem 3 ([, Generalized Popoviciu inequality])
where and for .
The above inequalities were generalized as follows.
Theorem 4 ()
where , , and for .
Definition 2 ()
holds for all and .
The following inequalities for s-convex functions were established.
Theorem 5 ([, Theorem 4.2])
Theorem 6 ([, Theorem 4.4])
where and for .
The concept of h-convex functions below was innovated as follows.
Definition 3 ([, Definition 4])
for all and .
Definition 4 ([, Section 3])
is valid for all . If the inequality (11) reverses, then f is said to be a sub-multiplicative function on J.
The following inequalities were established for .
Theorem 7 ([, Theorem 6])
where . If h is sub-multiplicative and , then the inequality (12) is reversed.
Theorem 8 ([, Theorem 11])
where . The inequality (13) is reversed if .
Theorem 9 ([, Theorem 12])
where and for and . The inequality (14) is reversed if .
Two new kinds of convex functions were introduced as follows.
Definition 5 ()
is valid for all and , then we say that is an m-convex function on .
Definition 6 ()
If the inequality (16) is reversed, then f is said to be -concave and denoted by .
The aim of this paper is to find some inequalities of Jensen type and Popoviciu type for -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 -convex functions.
If h is sub-multiplicative and , then the inequality (17) is reversed.
Proof Assume that for .
When , taking and in Definition 6 gives the inequality (17) clearly.
for . Namely, when , the inequality (17) holds. By induction, Theorem 10 is proved. □
- 1.if , we have(19)
- 2.if , we have(20)
if h is sub-multiplicative and , then the inequalities (19) and (20) are reversed.
If , then the inequality (22) is reversed.
If h is sub-multiplicative and , then the inequality (23) is reversed.
The proof of Theorem 11 is complete. □
is valid, where .
- 1.if , then(25)
- 2.if , then(26)
if h is sub-multiplicative and , then the inequalities (25) and (26) are reversed.
- 1.if for , then(27)
if , then the inequality (27) is reversed.
where , …, .
If h is sub-multiplicative and , then the inequality (28) is reversed.
The proof of Theorem 12 is complete. □
- 1.When , we have(31)
- 2.When and , we have(32)
- 3.When and , we have(33)
If h is sub-multiplicative and , then the inequalities (31) to (33) are reversed.
Remark 1 The inequality (14) can be deduced from applying (33) to for , , and for .
- 1.if for , then(34)
- 2.if for and , then(35)
- 3.if and , then(36)
if , then the inequalities (34) to (36) are reversed.
where , …, .
If h is sub-multiplicative and , then the inequality (37) is reversed.
The proof of Theorem 13 is complete. □
- 1.When , we have(39)
- 2.When and , we have(40)
- 3.When and , we have(41)
If h is sub-multiplicative and , then the inequalities (39) to (41) are reversed.
- 1.if for , then(42)
- 2.if and for , we have(43)
- 3.if and , then(44)
if , then the inequalities (42) to (44) 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.
- 1.if and , or if and , we have
- 2.if , , and , we have
if and , or if and , the function ;
if , , and , the function .
By virtue of Corollary 10, we obtain the following results.
- 1.If and , or if and , then we have(45)
- 2.if or and if , we have(46)
- 3.if or and if , then(47)
if , , and , then the inequality (47) are reversed.
- 1.If , we have(48)
- 2.if and , we have(49)
- 3.if and , then(50)
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.
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