Čebyšev inequalities for co-ordinated QC-convex and (s,QC) -convex

Čebyšev inequalities for co-ordinated QC-convex and (s, QC)-convex B. Meftah1,∗ and A. Souahi2 1 Laboratoire des Télécommunications, Faculté des Sciences et de la Technologie, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria. 2 Laboratory of Advanced Materials, University of Badji Mokhtar-Annaba, P.O. Box 12, 23000 Annaba, Algeria. * Correspondence: badrimeftah@yahoo.fr Received: 19 July 2020; Accepted: 1 January 2021; Published: 23 January 2021. Abstract: In this paper, we establish some new Čebyšev type inequalities for functions whose modulus of the mixed derivatives are co-ordinated quasi-convex and α-quasi-convex and s-quasi-convex functions.


Introduction
I n 1882,Čebyšev [1] gave the following inequality where f , g : [a, b] → R are absolutely continuous function, whose first derivatives f and g are bounded and and . ∞ denotes the norm in L ∞ [a, b] defined as f ∞ = ess sup t∈ [a,b] | f (t)|.
Recently, Guezane-Lakoud and Aissaoui [2] gave the analogue of the functional (2) for functions of two variables and established the followingČebyšev type inequalities for functions whose mixed derivatives are bounded as follows; and where Motivated by the existing results, in this paper we establish some newČebyšev type inequalities for functions whose mixed derivatives are co-ordinates quasi-convex and co-ordinates (α, QC) and (s, QC)-convex.

Main result
Theorem 1. Let f , g : ∆ → R be partially differentiable functions such that their second derivatives f λw and g λw are integrable on ∆. If | f λw | and |g λw | are co-ordinated quasi-convex on ∆, then where T( f , g) is defined as in (5), M = max and Multiplying (8) by (9), and then integrating the resulting equality with respect to x and y over ∆, using modulus and Fubini's Theorem, and multiplying the result by 1 k , we get Since | f λα | and |g λα | are co-ordinated quasi-convex, we deduce where we have used the fact that The proof is completed.

Theorem 2.
Under the assumptions of Theorem 1, we have where T( f , g) is defined as in (5), M, N, and k are as in Theorem 1.
Proof. From Lemma 1, (8) and (9) are valid. Let G(x, y) = 1 2k g(x, y) and F(x, y) = 1 2k f (x, y). Multiplying G(x, y) by F(x, y), then integrating the resultant equalities with respect to x and y over ∆, and by using the modulus, we get Since | f λw | and |g λw | are co-ordinated quasi-convex, (14) implies By a simple computation, we easily obtain Substituting (16) in (15), we get the desired result.
Theorem 3. Let f , g : ∆ → R be partially differentiable functions, such that their second derivatives f λw and g λw are integrable on ∆. If | f λw | and |g λw | are co-ordinated α-quasi-convex on ∆, for some α ∈ (0, 1], then where T( f , g) is defined as in (5), M, N, and k are as in Theorem 1.

Theorem 4. Under the assumptions of Theorem 3, we have
where T( f , g) is defined as in (5) and M, N, and k are as in Theorem 3.
Proof. By the same argument given in Theorem 2, we easly obtain the inequality (14), using the α-quasi-convexity on the co-ordinates of | f λw | and |g λw |, we get Substituting (16) in (20), we get the desired result.
Theorem 5. Let f , g : ∆ → R be partially differentiable functions such that their second derivatives f λw and g λw are integrable on ∆, and let s ∈ (−1, 1] fixed. If | f λα | and |g λα | are co-ordinated s-quasi-convex on ∆, then where T( f , g) is defined as in (5) and M, N, and k are as in Theorem 1.

Theorem 6.
Under the assumptions of Theorem 5, we have where T( f , g) is defined as in (5) and M, N, and k are as in Theorem 1.
Proof. By the same argument given in Theorem 2, we easily obtain the inequality (14), using the second definition of s-quasi-convexity on the co-ordinates of | f λw | and |g λw |, we get |T( f , g)| ≤ 1 Substituting (16) in (24), we get the desired result.
Author Contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflicts of Interest: "The authors declare no conflict of interest."