Abstract

In the article, we establish several Petrović-type inequalities for the harmonic -convex (concave) function if is a submultiplicative (super-multiplicative) function, provide some new majorizaton type inequalities for harmonic convex function, and prove the superadditivity, subadditivity, linearity, and monotonicity properties for the functionals derived from the Petrović type inequalities.

1. Introduction

Let be a nonempty interval. Then a real-valued function is said to be convex (concave) on if the inequality

holds for all and .

It is well known that the convex (concave) functions have wide applications in pure and applied mathematics [1–21], many remarkable properties and inequalities can be found in the literature [22–48] via the theory of convexity. Recently, a great deal of generalizations, extensions, and variants have been made for convexity, for example, GA-convexity [49], GG-convexity [50], -convexity [51, 52], preinvex convexity [53], strong convexity [54–57], Schur convexity [58–60], and others [61–67].

The following well-known Petrović inequality was established by Petrović [68] in 1932.

Theorem 1. Let and be a convex function. Then the inequalityholds for all nonnegative -tuples and such that .

Now, we recall the definitions of various convexities and the majorization relation between two -tuples.

Definition 2. Let be an interval. Then a real-valued function is said to be submultiplicative (supermultiplicative) on if the inequalityholds for all .

Definition 3. Let , and be two -tuples. Then is said to be majorized by or majorizes (in symbol ) iffor andwhere denotes the th largest component in .

Definition 4. Let be an interval. Then a real-valued function is said to be harmonic convex (concave) on if the inequalityholds for all with and .

Definition 5. Let , , and be an interval. Then a real-valued function is said to be harmonic -convex (concave) on if the inequalityholds for all with and .

Definition 6. Let and be an interval. Then a real-valued function is said to be harmonic -convex (concave) in the second sense if the inequalityholds for all with and .

Definition 7. Let , , and be an interval. Then a real-valued function is said to be harmonic -convex (concave) on if the inequalityholds for all and with and .

Definition 8. Let be an interval and be a nonnegative real-valued function. Then a real-valued function is said to be harmonic -convex (concave) if the inequality

holds for all and with .

Abbas Baloch [69] proved that.

Lemma 9. Let and be a real-valued function. Then is harmonic convex on if and only if the function is convex on .

The following three examples also can be found in the literature [67].

Example 1. Let . Then is harmonic convex on and harmonic concave on .

Example 2. Let be defined byThen is harmonic convex on due to is convex on .

Example 3. The functionis harmonic convex on due to is convex on .

Lemma 10. (see [70]). Let be an interval and be a real-valued function. Then is convex on if and only if the bivariate functionis increasing with respect to its variables and .

2. Main Results

Lemma 11. Let with , and be two n-tuples such thatfor , andandbe two real-valued functions. Then one hasif the function is increasing on .

Proof. It follows from (14) and the monotonicity of the function on the interval thatfor all .
From above inequality we clearly see that

Theorem 12. Let with , , and be two n-tuples such thatfor , and be a supermultiplicative function such thatfor all . Then one hasif is a harmonic -convex function.

Proof. It follows from Lemma 1 and the harmonic -convexity of the function on that the function is -convex on . Let such that andThen we clearly see that andSince is a supermultiplicative function, we haveMaking use of (19) we getwhich implies that the function is increasing on . Then from Lemma 11 one haswhich is equivalent toTherefore, inequality (20) can be obtained immediately by replacing in the last inequality above.
Let . Then is a supermultiplicative function and inequality (19) becomes an identity, and Theorem 12 leads to Theorem 13 immediately.

Theorem 13. Let with , , and be two n-tuples such thatfor . Then the inequalityholds if is a harmonic convex function.

Remark 14. Let with , , and be two n-tuples such thatfor . Then from the proof of Theorem 12 we clearly see that the following statements are true:(1)If and are two functions such that is decreasing on , then the reverse inequality of (15) holds.(2)If is a submultiplicative function such that for all and is a harmonic -concave function, then the reverse inequality of (20) holds.(3)If is a harmonic concave function, then the reverse inequality of (28) holds.

Theorem 15. Let , , , such that , and be a harmonic convex function. Then the inequalityholds.

Proof. Let andThen from Lemmas 1 and 2 together with the harmonic convexity of that is convex on and the sequence is decreasing. LetThen due to andTherefore, inequality (4.0) follows from the monotonicity of the sequence and for all together with the last identity above.
The weighted version of Theorem 15 can be stated as follows.

Theorem 16. Let , , , such that , and be a harmonic convex function. Then the inequalityholds.

In this section, we present several interesting properties for the functionals derived from the Petrović-type inequalities given in Section 2.

Let with , , and be two n-tuples such that

for , and and be two real-valued functions. Then the Petrović-type functional is defined by

Remark 17. From Lemma 11 we clearly see thatif is increasing on andif is decreasing on .
Let , , , and be a real-valued function. Then the Petrović-type functional is defined by

Remark 18. Theorem 16 leads to the conclusion thatif and is harmonic convex on andif and is harmonic concave on .

Theorem 19. Let with , and be two n-tuples such thatfor , and and be two real-valued functions, and be defined by (36). Then the following statements are true:(1)If and is increasing on , thenthat is is superadditive on .(2)If (namely, for all ) and for all , and is increasing on , thenthat is is increasing with respect to positive n-tuples.(3)If and is decreasing on , then the reverse inequality of (43) holds, that is is subadditive on . If and for all, andis decreasing on, then the reverse inequality of (44) holds, that isis decreasing with respect to positiven n-tuples.

Proof. (1) It follows from (36) thatMaking use of the monotonicity of the function and Lemma 11 one hasCombining (45) and (46) givesTherefore,follows from (47).
(2) It follows from part (1) and Remark 17 that(3) Part (3) can be proved by similar methods as in proving parts (1) and (2).

Theorem 20. Let , with , , be a harmonic convex function, and be defined by (39). Then the following statements are true:(1) is linear with respect to , that is(2) If , thenthat is is increasing with respect to .(3) If is harmonic concave on , then inequalities (50) and (51) also hold true.

Proof. (1) It follows from (39) that(2) From part (1) and Remark 18 we get(3) Part (3) can be proved by similar methods in proving parts (1) and (2).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 61673169, 11301127, 11701176, 11626101, 11601485) and the Higher Education Commission of Pakistan.