Abstract

In this paper, a well-known inequality called Giaccardi inequality is established for isotonic linear functionals by applying s-convexity in the second sense, which leads to notable Petrović inequality. As a special case, discrete and integral versions of Giaccardi inequality are derived along with the Petrović inequality as a particular case. In application point of view, newly established inequalities are derived for different time scales.

1. Introduction

Many important areas of mathematics have been developed just because of convex functions. Due to important properties and characterizations of convex functions supreme branches of pure and applied mathematics are studied extensively. A convex function in is a function, whose epigraph is a convex set. A convex function is always continuous and has left and right derivatives in the interior of its domain. More generally, a function is convex on an interval if for any two points and of and any , where , it satisfies the following inequality:

Many researchers had generalized the notion of convex functions by inducting different parameters and functions provided the inequality (1) must retain preserved. For instance -convex functions [1], -convex functions [2], -convex functions [3] and -convex functions [4, 5] can be seen in this regard. One of the important generalizations of convex functions is -convex functions [6]. The -convex function in the second sense was introduced by H. Hudzik and L. Maligranda in [1].

Definition 1. Let and be an interval. A function is said to be -convex in the second sense iffor all and with .
The class of -convex functions in the second sense is usually denoted by .

Example 1 (see [7]). Let and . Define the function as follows:It can be easily checked.(i)if and , then (ii)if and , then .There are many publications in the literature that explore this form of convexity. Theories and inequalities underlying -convexity by analytical methods are currently being discussed. It can be easily seen that in case , -convexity is reduced to the ordinary convexity of function .
Let us recall the definition of isotonic linear functionals from [8].

Definition 2. Letbe a nonempty set andbe a class of real-valued functions defined on having the properties:L1: if , then for all L2: , that is, if for , then .An isotonic linear functional is a functional that satisfies the following axioms:A1: for A2: for .Isotonic linear functionals contribute in a wide area of pure and applied mathematics and play a key role in resolving many problems and inequalities. One can note the common examples of such isotonic functionals are given in [8] as follows:where is a positive measure on some suitable set .
Giaccardi inequality for isotonic linear functionals for convex functions is stated in the following theorem.

Theorem 1 (see [8]). Let, satisfy the conditions L1, L2, and A1, A2. Also, let and such that and either for all or for all . If is convex on (or on ) and , then the following is valid:

The aim of this paper is to prove the Giaccardi inequality for -convex functions in the second sense for isotonic linear functionals. The well-known Petrović inequality is derived as a particular case of the newly established Giaccardi inequality. As a special case, discrete and integral versions of these inequalities are derived. As applications on the time scale, Giaccardi and Petrović inequalities are discussed for different time scales.

2. Main Results

In the following theorem, we prove Giaccardi inequality for -convex functions in the second sense for isotonic linear functionals.

Theorem 2. Let ,satisfy the conditions L1, L2, and A1, A2, respectively. Also, let such that and eitherIf is -convex in second sense on or on , and , then,where

Proof. Let and be points from the interval having end points and such that . It is easy to check thatIf we take and , then . As is given to be -convex in second sense so using the value of and along with the values of and defined above in inequality (2), one hasSubstituting , and in above inequality to getNow setting and (or) and to getThat is,Applying the property A2 of the isotonic linear functional , one hasUsing the linearity of the functional , one getA simplification of the inequality (15) leads us to the required result.

Remark 1. If we take in Theorem 2, then one gets Giaccardi’s inequality for isotonic linear functional stated in [8] (as given in Theorem1).
In the following, a discrete version of the above theorem has been derived.

Theorem 3. Let, where is an interval, be positive numbers and such thatwhere . If is -convex in second sense on , thenwhere

Proof. Let , andThen, satisfies conditions L1, L2 and satisfies conditions A1, A2 of Definition 2. Substituting above values of and in Theorem 2, we getFinally, inequality (7) becomes our required result.
The following result provides us the integral version of Theorem 2.

Theorem 4. Let be a measurable space, where is positive finite measure. Also let be a measurable function and such that andIf is -convex on second sense on , then an inequalityis valid, provided the integrals exist and

Proof. Assume thatThen, satisfies conditions L1, L2 and satisfies conditions A1, A2 of Definition 1. Substituting above values of in Theorem 2, we getUltimately, inequality (7) becomes (22) as our required result.
In the following theorem, Petrović’s inequality for -convex functions in second sense for isotonic linear functionals is derived.

Theorem 5. Let be -convex in second sense on or on , then the inequality, holds for either Thwor all or for all. .

Proof. Put in (7) to get (26).

Remark 2. If we putin Theorem 5, one get Petrović’ inequality for isotonic linear functionals stated in [8].
The famous Giaccardi’s inequality given in [9] can be deduced directly from Theorem 2 as stated in the following corollary.

Corollary 1. Let , where is an interval, be positive numbers and such that condition (16) is valid. If is a convex on

Proof. Take in (17) to get (27).
In the following corollary, Petrović’s inequality for convex functions has been deduced.

Corollary 2. Let, be positive numbers such that, where . If is -convex in second sense on (or) , then

Proof. Put in (17) to get (29).
In the following corollary, Petrović’s inequality [9] has been derived.

Corollary 3. Let ,be positive numbers such that condition (28) is valid. Ifis convex on, then

Proof. Put and in (17) to get (30).

Corollary 4. Letbe a measurable space, whereis a positive finite measure. Also letbe a measurable function andsuch thatand condition (21) is valid. If is convex on , then the following is valid:Provided the integrals exist.

Proof. It is a simple consequence of Theorem 4 for .

Corollary 5. Let be a measurable space, where is positive finite measure. Also let be a measurable function andsuch that andIf is -convex in second sense on , then the following is valid:Provided the integrals exist.

Proof. Put in (22) to get (33).
The following result holds as the integral version of the famous Petrović’s inequality.

Corollary 6. Let be a measurable space, whereis a positive finite measure. Also letbe a measurable function andsuch that (32) is valid. If is a convex on , then the following is valid:Provided the integrals exist.

Proof. Put in Corollary 5 to get the required result.

3. Applications in Time Scale Calculus

The theory of time scale was introduced by S. Hilger in his Ph.D. Thesis in 1988 (see [10]). A time scale is an arbitrary nonempty closed subset of the real numbers , and we usually denote it by the symbol . The real numbers, the integers, the nonnegative integers, and the natural numbers are examples of time scales. In this section, we give Giaccardi and Petrović inequalities for -convex functions in the second sense for isotonic linear functional defined on different time-scales taken from the literature (for example, see [11]).(a)For with , suppose consists of all real-valued functions defined on and . Let such that the assumptions of Theorem 2 are satisfied. If is an -convex in second sense on (or ) such that , thenUnder the same conditions, Petrović’s inequality for -convex in second sense for time-scale can be derived by putting in (35), that is,(b)For and with , suppose consists of all real-valued functions defined on and . Also let such that the assumptions of Theorem 2 are satisfied. If is -convex in second sense on (or ) such that , thenUnder the same conditions, Petrović’s inequality for -convex functions in second sense for time-scale can be derived by putting in (37), that is,(c)For , with , supposeAlso, let such that the assumptions of Theorem 2 are satisfied. If is -convex in second sense on (or ) such that , thenUnder the same conditions, Petrović’s inequality for -convex functions in the second sense for time-scale can be derived by putting in (40), that is,(d)For , with , supposewhere represents rd-continuous functions [12] and the integral is the Cauchy delta time-scale integral [13]. Also, let such that the assumptions of Theorem 2 are satisfied. If is -convex in second sense on (or ) such that , thenUnder the same conditions, Petrović’s inequality for -convex functions in second sense for time-scale can be derived by putting in (43), that is,(e)For , with , supposewhere represents ld-continuous functions [14] and the integral is the Cauchy nabla time-scale integral [15]. Also, let such that the assumptions of Theorem 2 are satisfied. If is -convex in second sense on (or ) such that , then,Under the same assumptions, Petrović’s inequality for -convex functions in second sense for time-scale can be derived by putting in (46), that is,(f)For , with , supposewhere the integral is the Cauchy -diamond time-scale integral [16]. Also, let such that the assumptions of Theorem 2 are satisfied. If is -convex in second sense on (or ) such that , thenUnder the same assumptions, Petrović’s inequality for -convex functions in second sense for given time-scale can be derived by putting in (49), that is,(g)Assume that , , …, are time scales and , such that , for . Also, supposebe Jordan -measurable and let be the set of all bounded -integrable functions from to . Moreover, letwhere the integral is the multiple Riemann delta time-scale integral [11]. Also, let such that the assumptions of Theorem 2 are satisfied. If is -convex in second sense on (or ) such that , thenUnder the same assumptions, Petrović’s inequality for -convex functions in second sense for a given time-scale that can be derived by putting in (53), that is(h)Under the time-scale defined in the above-given clause 7, let be Lebesgue -measurable, be the set of all -integrable functions from to and , for , be multiple Lebesgue delta time-scale integral (see [17]). If the assumptions of Theorem 2 are satisfied under these conditions, then for -convex function in the second sense on (or , the inequalities (53) and (54) are valid.

4. Discussion

The most famous Giaccardi and Petrović inequalities for -convex functions in the second sense for isotonic linear functionals are derived. It has been shown that these new findings generate many classical results for different particular cases of the isotonic linear functionals. While understanding the significance of time-scale calculus, these inequalities have been derived for various time-scale integrals. It is important to note that the results derived in this paper for isotonic linear functional can be particularized for other branches of science.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research work was supported by Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province under Grant no. MSSB-2021-14. The research work of Atiq ur Rehman is partially supported by the Higher Education Commission of Pakistan with NRPU no. 7962.