Ansari et al. Advances in Difference Equations https://doi.org/10.1186/s13662-020-03159-x (2020) 2020:698 RESEARCH Open Access Some inequalities for Csiszár divergence via theory of time scales Iqrar Ansari1* , Khuram Ali Khan1 , Ammara Nosheen2 , Ðilda Pečarić3 and Josip Pečarić4 * Correspondence: iqrar@math.qau.edu.pk 1 Department of Mathematics, University of Sargodha, Sargodha, Pakistan Full list of author information is available at the end of the article Abstract In this paper, we present some inequalities for Csiszár f -divergence between two probability measures on time scale. These results extend some known results in the literature and offer new results in h-discrete calculus and quantum calculus. We also present several inequalities for divergence measures. Keywords: Csiszár divergence; Time scales calculus; Quantum calculus; Convex function 1 Introduction In many applications of probability theory the essential problem is determining an appropriate measure of distance (or divergence) among two probability distributions. Consequently, many different divergence measures were introduced and extensively studied by various authors, for instance, the Csiszár f -divergence (Kullback–Leibler divergence, Hellinger distance, and total-variation distance), Rényi divergence, and Jensen–Shannon divergence; see [9, 13, 18, 20]. Csiszár [6] introduced the following: Definition 1 Let f : R+ → R+ be a convex function. Let r̃ = (r1 , r2 , . . . , rn ) and s̃ =   (s1 , s2 , . . . , sn ) be such that nν=1 rν = 1 and nν=1 sν = 1. Then the f -divergence functional is defined as  rν If (r̃, s̃) := , sν f sν ν=1 n   where f satisfies the following conditions: f (0) := lim+ f (θ ); θ→0   0 := 0; 0f 0     a a 0f := lim+ θ f , θ→0 0 0 a > 0. Dragomir [7, 8] has done a plenty of work giving different types of bounds on the distance and divergence measures. Jensen’s inequality plays a vital role to get inequalities for divergences between probability distributions. Horvath et al. [11] introduced a new © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Ansari et al. Advances in Difference Equations (2020) 2020:698 functional based on the f -divergence functional and obtained some estimates for the new functional, the f -divergence and Rényi divergence by utilizing cyclic refinement of Jensen’s inequality. Recently, Adil et al. [12] obtained some inequalities for convex functions and their applications to Csiszár divergence. The main objective behind the theory of time scales is unifying continuous and discrete analysis introduced by Stefan Hilger in 1988 and established in the comprehensive books [4, 5]. Various dynamic derivatives on time scales not just give a helpful route in useful applications, but also demonstrate their extraordinary appearance in approximations. It may be beneficial to examine if such useful features can be kept up or even improved in a specific way while different dynamic derivatives are utilized in the same application simultaneously. Guseinov [10] examined the process of Riemann and Lebesgue integration on time scales. Many authors established time scale version of linear and nonlinear integral inequalities [1, 17, 19]. The time scale integral inequalities have been used to study the boundedness, uniqueness, and so on of the solutions of different dynamic equations [14, 16]. Ansari et al. [2] introduced the differential entropy of a continuous random variable on time scales and established some Shannon-type inequalities on arbitrary time scales. It was shown that the obtained inequalities are used to estimate the bounds of differential entropy for some particular distributions. Some classical inequalities and their converses for multiple integration on time scales were investigated in [3]. The setup of this paper is as follows. Section 2 is confined to the basic definitions and preliminary results of time scales calculus. Our aim in Sect. 3 is deriving some new inequalities for Csiszár f -divergence on arbitrary time scales and finding some inequalities for Csiszár divergence in h-discrete calculus and quantum calculus. To the best of the author’s knowledge, no contribution is available in the literature for Csiszár divergence inequalities in quantum calculus. Section 4 is concerned to the study of some divergence measures on time scales including the bounds of the Kullback–Leibler distance, triangular discrimination, Hellinger discrimination, Jeffreys distance, Bhattacharyya distance, and harmonic distance in terms of some special means such as identric, logarithmic, arithmetic, and geometric means. The upper bounds of these divergence results in quantum calculus are also part of discussion. 2 Preliminaries In this paper, we assume that a time scale T is an arbitrary nonempty closed subset of the real line. The following definitions and results are extracted from [4]. Definition 2 Consider a time scale T that is a closed and bounded subset of real numbers and ω ∈ T. Then the mappings σ : T → T and ρ : T → T satisfying σ (ω) = inf{λ ∈ T : λ > ω} and ρ(ω) = sup{λ ∈ T : λ < ω} are known as forward and backward jump operators on T, respectively. A function z : T → R is right-dense continuous or rd-continuous if it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T. The set of all rd-continuous functions is denoted by Crd . Page 2 of 21 Ansari et al. Advances in Difference Equations (2020) 2020:698 Page 3 of 21 Here we define Tk as follows: ⎧ ⎨T\(ρ(sup T), sup T] if sup T < ∞, Tk = ⎩T if sup T = ∞. Definition 3 Let z : T → R and ω ∈ Tk . Then we define the delta derivative z (ω) as the number (provided it exists) such that for each > 0, there exists a neighborhood U of ω such that z σ (ω) – z(λ) – z (ω) σ (ω) – λ ≤ σ (ω) – λ for all λ ∈ U. We say that z is delta differentiable at ω. If T = R, then z is the the usual derivative z , whereas z becomes the forward difference operator z(ω) = z(ω + 1) – z(ω) for T = Z. If T = qZ = {qn : n ∈ Z} ∪ {0} with q > 1, then z is the so-called q-difference operator z (ω) = z(qω) – z(ω) , (q – 1)ω z (0) = lim λ→0 z(λ) – z(0) . λ Theorem 1 (Existence of antiderivatives) Every rd-continuous function has an antiderivative. In particular if x0 ∈ T, then F is defined by x f (ω)ω F(x) := for x ∈ Tk x0 which is an antiderivative of f . b For T = R, we get a z(ω)ω = where a, b ∈ T with a ≤ b. b a z(ω) dω, and if T = N, then b a z(ω)ω = b–1 ω=a z(ω), 3 Main results Let T be a time scale and consider the set of all probability density functions on T,  := r̃ ∈ Crd [a, b]T , [0, ∞) , r̃(x) ≥ 0, b  r̃(x)x = 1 . a In this paper, we assume that r̃, s̃ ∈ . Definition 4 The Csiszár f -divergence on time scales is defined as b Df (s̃, r̃) := a   s̃(x) r̃(x)f x, r̃(x) where f is convex on (0, ∞). By suitable substitutions to f in Definition 4 we can obtain several divergences on time scales. For instance, if we choose f (x) = x2 – 1, then we find the Pearson χ 2 -divergence on Ansari et al. Advances in Difference Equations (2020) 2020:698 Page 4 of 21 tim