Generalized Jensen-Mercer Inequality for Functions with Nondecreasing Increments

and Applied Analysis 3 (i) A function with nondecreasing increments is not necessarily continuous. (ii) If the first partial derivatives of a function f : U → R exist for x ∈ U, then x has nondecreasing increments if and only if each of these partial derivatives is nondecreasing in each argument. (iii) If the second partial derivatives of a functionf : U → R exist for x ∈ U, then x has nondecreasing increments if and only if each of these partial derivatives is nonnegative. (iv) If a function f with nondecreasing increments is continuous for b ≤ x ≤ a + b, where 0 ≤ a ∈ R, then the functionφ : [0, 1] → Rdefined byφ(t) = f(ta+b) is convex. We define here a special type of functions which belong to the class of functions with nondecreasing increments and which themselves contain the class of convex functions.These functions are called Wright convex functions [1, page 7]. Definition 7. We say f : I → R is a Wright convex function if ∀a, b + h ∈ I with a < b and h > 0 we have f (a + h) − f (a) ≤ f (b + h) − f (b) . (12) Remark 8. It is easy to see that, in one-dimensional case, functions with nondecreasing increments are Wright convex functions. Also, continuous Wright convex functions are convex functions. Thus, the class of convex functions is a proper subclass of the Wright convex functions. Now we state some results that will be needed to derive our main results. The following proposition gives Jensen’s inequality for functions with nondecreasing increments [11]. Proposition 9. Let f : U → R be a continuous function with nondecreasing increments, let w be a nonnegative n-tuple such thatWn > 0, and let x ∈ U, where i ∈ {1, . . . , n}, be such that x ≤ ⋅ ⋅ ⋅ ≤ x or x ≥ ⋅ ⋅ ⋅ ≥ x. Then, it holds that f( 1 Wn n ∑ i=1 wix (i) ) ≤ 1 Wn n ∑ i=1 wif (x (i) ) . (13) We now state Jensen-Steffensen’s inequality for functions with nondecreasing increments [12]. Proposition 10. f : [a, b] → J is a nondecreasing continuous function andH : [a, b] → R is of bounded variation satisfying H(a) ≤ H (t) ≤ H (b) ∀t ∈ [a, b] , H (b) > H (a) . (14) If φ : J → R is a continuous function with nondecreasing increments, then the following inequality holds: φ( ∫ b a f (t) dH (t) ∫ b a dH (t) ) ≤ ∫ b a φ (f (t)) dH (t) ∫ b a dH (t) , (15) where ∫b a fdH = (∫b a f1dH, . . . , ∫ b a fkdH). At this stage, we prove Jensen-Boas inequality for functions with nondecreasing increments as follows. Theorem 11. Let f : [a, b] → J be a continuous andmonotonic (either nonincreasing or nondecreasing) map in each of the k intervals (bi−1, bi). Let H : [a, b] → R be continuous or of bounded variation satisfying H(a) ≤ H (a1) ≤ H (b1) ≤ H (a2) ≤ ⋅ ⋅ ⋅ ≤ H (bk−1) ≤ H (ak) ≤ H (b) (16) for all ai ∈ (bi−1, bi) (b0 = a, bk = b) and H(b) > H(a). If φ is a continuous function having nondecreasing increments in each of the k intervals (bi−1, bi), then we have the following inequality: φ( ∫ b a f (t) dH (t) ∫ b a dH (t) ) ≤ ∫ b a φ (f (t)) dH (t) ∫ b a dH (t) . (17) Proof. Using Jensen’s inequality (13) for nonnegative n-tuple w and Jensen-Steffensen’s inequality (15), if H(a) ≤ H(a1) ≤ H(b1) ≤ H(a2) ≤ ⋅ ⋅ ⋅ ≤ H(bk−1) ≤ H(b) we have φ( ∫ bi bi−1 f (t) dH (t) ∫ bi bi−1 dH (t) ) ≤ ∫ bi bi−1 φ (f (t)) dH (t) ∫ bi bi−1 dH (t) (18) for i ∈ {1, 2, . . . , k}. If we consider ti = (∫ bi bi−1 f(t)dH(t)/ ∫b bi−1 dH(t)) and wi = ∫ bi bi−1 dH(t), then we can write φ (ti) ≤ 1 wi ∫ bi bi−1 φ (f (t)) dH (t) ; (19) using this fact, we have φ( ∫ b a f (t) dH (t) ∫ b a dH (t) ) = φ( ∑ n i=1 witi ∑ n i=1 wi ) ≤ ∑ n i=1 wiφ (ti) ∑ n i=1 wi ≤ ∑ n i=1 wi (1/wi) ∫ bi bi−1 φ (f (t)) dH (t) ∑ n i=1 wi = ∫ b a φ (f (t)) dH (t) ∫ b a dH (t) . (20) The following proposition represents an integral majorization result which would be needed in our next main result [1, page 328]. 4 Abstract and Applied Analysis Proposition 12. Let f , g : [a, b] → J be two nonincreasing continuous functions and let H : [a, b] → R be a function of bounded variation. If


Introduction and Preliminaries
Let us start with Jensen's inequality for convex functions, one of the most celebrated inequalities in mathematics and statistics (for detailed discussion and history, see [1,2]).Throughout the paper, we assume that  and [, ] are intervals in R, and J is an interval in R  and U ⊂ R  is a -dimensional rectangle for integer  ≥ 1.Also for weights   ,  ∈ {1, . . ., }, we would use   = ∑  =1   .
In [3], McD Mercer proved the following variant of Jensen's inequality, which we will refer to as Mercer's inequality.
Proposition 2. Under the assumptions of Proposition 1, the following inequality holds: where There are many versions, variants, and generalizations of Propositions 1 and 2; see for example [4][5][6][7].Here we state an integral version of Jensen's inequality from [1, pages 58-59] which will be needed in our main result. holds.
In our construction for next proposition, we recall the definitions of majorization.
when x ≺ y, x is said to be majorized by y or y majorizes x.
This notion and notation of majorization were introduced by Hardy et al. in [8].
If  majorizes each row of X, that is, then we have the inequality where   > 0 with nonnegative weights   .
The present paper is organized as follows: after some preliminaries, in Section 2, we recall definition of functions with nondecreasing increments and their properties and note that some inequalities from Section 1 which held true for convex functions also hold for functions with nondecreasing increments.In Section 3, we give an integral generalization of Niezgoda's inequality.In the process, we will use an integral majorization result of Pečarić [9] and prove a result which gives the Jensen-Boas inequality on disjoint set of subintervals for functions with nondecreasing increments.In Section 4, we will discuss some refinements of the main results we proved in Section 3. The last part of this section is devoted to the applications of some related results.

Introduction to Functions with Nondecreasing Increments
In 1964, Brunk defined an interesting class of multivariate real valued functions [10] known as functions with nondecreasing increments.
In the same paper [10], Brunk gave some examples and properties of the functions which we discuss below.

Examples of Functions with Nondecreasing Increments
(i) The simplest example of a function with nondecreasing increments is a constant function.
(ii) Lines of the form x = a + b, where (0, . . ., 0) ≤ a ∈ R  and b ∈ R  whose direction cosines are nonnegative, also belong to the family of functions with nondecreasing increments.
(iv) An interesting and widely used example of such functions is the Cauchy functional equation

Properties of Functions with Nondecreasing Increments.
Functions with nondecreasing increments possess the following properties: (i) A function with nondecreasing increments is not necessarily continuous.
(ii) If the first partial derivatives of a function  : U → R exist for x ∈ U, then x has nondecreasing increments if and only if each of these partial derivatives is nondecreasing in each argument.
(iii) If the second partial derivatives of a function  : U → R exist for x ∈ U, then x has nondecreasing increments if and only if each of these partial derivatives is nonnegative.
(iv) If a function  with nondecreasing increments is continuous for b ≤ x ≤ a + b, where 0 ≤ a ∈ R  , then the function  : [0, 1] → R defined by () = (a+b) is convex.
We define here a special type of functions which belong to the class of functions with nondecreasing increments and which themselves contain the class of convex functions.These functions are called Wright convex functions [1, page 7].Definition 7. We say  :  → R is a Wright convex function if ∀,  + ℎ ∈  with  <  and ℎ > 0 we have Remark 8.It is easy to see that, in one-dimensional case, functions with nondecreasing increments are Wright convex functions.Also, continuous Wright convex functions are convex functions.Thus, the class of convex functions is a proper subclass of the Wright convex functions.Now we state some results that will be needed to derive our main results.The following proposition gives Jensen's inequality for functions with nondecreasing increments [11].Proposition 9. Let  : U → R be a continuous function with nondecreasing increments, let w be a nonnegative n-tuple such that   > 0, and let x () ∈ U, where  ∈ {1, . . ., }, be such that x (1) ≤ ⋅ ⋅ ⋅ ≤ x () or x (1) ≥ ⋅ ⋅ ⋅ ≥ x () .Then, it holds that We now state Jensen-Steffensen's inequality for functions with nondecreasing increments [12].If  : J → R is a continuous function with nondecreasing increments, then the following inequality holds: where At this stage, we prove Jensen-Boas inequality for functions with nondecreasing increments as follows.
If we consider   = (∫ (), then we can write using this fact, we have The following proposition represents an integral majorization result which would be needed in our next main result [1, page 328].

Proposition 12. Let f, g : [𝑎, 𝑏] → J be two nonincreasing continuous functions and let 𝐻 : [𝑎, 𝑏] → R be a function of bounded variation. If
hold, then for every continuous function with nondecreasing increments  : J → R the following inequality holds: Remark 13.If f, g : [, ] → J are two nondecreasing continuous functions such that then again inequality (22) holds.In this paper, we will state our results for nonincreasing f and g satisfying the assumption of Proposition 12, but they are still valid for nondecreasing f and g satisfying the above condition (see, e.g., [13, page 584]).

Generalized Jensen-Mercer Inequality
Here we state a result needed in the main theorems of this section.The following lemma shows that the subintervals in the Jensen-Boas inequality (see Theorem 11) can be disjoint for the inequality of type (15) to hold.

Lemma 14. Let 𝐻 : [𝑎, 𝑏] → R be continuous or a function of bounded variation and let
then, for every function f : [, ] → J which is continuous and monotonic (either nonincreasing or nondecreasing) in each of the  intervals (  ,   ) and every continuous function with nondecreasing increments  : J → R, the following inequality holds: Proof.Denote   = ∫     ().Due to (16), if (  ) = (  ) then  is a null-measure on [  ,   ] and   = 0, while otherwise   > 0. Denote  = { :   > 0} and Notice that and, due to Proposition 10, Therefore, taking into account the discrete Jensen's inequality (13), The following theorem is our main result of this section and it gives a generalization of Proposition 5.
Furthermore, let (, Σ, ) be a measure space with positive finite measure , let g : [, ] → J be a nonincreasing continuous function, and let f :  × [, ] → J be a measurable function such that the mapping   → f(, ) is nonincreasing and continuous for each  ∈ : Then, for a continuous function with nondecreasing increments  : J → R, the following inequality holds: Proof.Using Fubini's theorem, inequality (30), and Jensen's integral inequality (4), we have Applying Lemma 14 and Proposition 12, respectively, we have The special case of Theorem 15 can be found in [14] which may be stated as follows.

Refinements
Let (, Σ, ) be a measure space with positive finite measure .Throughout this section, we assume that  ⊂  with (), (  ) > 0 and we use the following notations: The following refinement of (31) is valid.
Theorem 17.Under the assumptions of Theorem 15, the following refinement is valid for every continuous function with nondecreasing increments  : J → R: Proof.Using discrete Jensen's inequality (13) for functions with nondecreasing increments, we have for any , which proves the first inequality in (37).
By inequality (31), we also have for any , which proves the second inequality in (37).(42) The special case of Theorem 17 can be found in [14] which may be stated as follows.(44) 4.1.Applications.Haluška and Hutník discussed a class of generalized weighted quasiarithmetic means in the integral form  [,], (, ) using the integral form of Jensen's inequality [15].In their work, they used the definition of quasiarithmetic nonsymmetrical weighted mean proposed by Feng [16] where  −1 denotes the inverse of the function .
In what follows,  is always a real continuous and strictly monotone function (in accordance with Definition 20).Means  [,], (, ) include many commonly used twovariable integral means as particular cases when taking the suitable functions , , and .For instance, (48) The case  = 0 corresponds to the weighted geometric mean.
Under the assumptions of Corollary 19, we define the following notations where  ∈ {, ,   }.Throughout this section, we also assume that ln and exp have the natural domain.
Arithmetic Mean.It is as follows: Geometric Mean.It is as follows: Harmonic Mean.It is as follows: Power Mean.It is as follows: We now define a relationship between arithmetic and geometric means.
Proof.In (43), let () = − ln() to get In our notation, we have Further simplification gives us Using the property of ln gives us which can be written as Finally, Here we obtain another relationship between geometric and arithmetic means.(60) In out notations, we have Finally, The following theorem states a relationship between geometric and harmonic means.)  ()  () . (63) In our notations, we have which can be written as Using the property of ln, we have Simplifying the above, we get Finally, we get Now we define another relationship between geometric and harmonic means. Finally, Now we define a relationship between power mean and arithmetic mean.