Abstract
In the article, we provide an example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex and its converse is not true in general, define the coordinate \((\eta _{1}, \eta _{2})\)-convex function and establish its Hermite–Hadamard type inequality.
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1 Introduction
Let \(I\subseteq \mathbb{R}\) be an interval. Then a real-valued function \(\varPsi : I\mapsto \mathbb{R}\) is said to be convex on I if the inequality
holds for all \(a, b\in I\) and \(\lambda \in (0, 1)\). Ψ is said to be concave if inequality (1.1) is reversed.
It is well known that the convexity theory has wide applications in special functions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], differential equations [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61] and bivariate means [62,63,64,65,66,67]. Recently, the extensions, generalizations, refinements and variants for the convexity have attracted the attention of many researchers. For example, Schur convexity [68,69,70], GA-convexity [71], GG-convexity [72], s-convexity [73, 74], preinvexity [75], strong convexity [76,77,78,79] and others [80,81,82,83,84,85].
Dragomir [86] defined the coordinate convex as follows.
Definition 1.1
(See [86])
Let \(I_{1}, I_{2}\subseteq \mathbb{R}\) be two interval, \(\varPsi : I_{1}\times I_{2}\mapsto \mathbb{R}\) be a real-valued function, and the partial mappings \(\varPsi _{y}: I_{1}\mapsto \mathbb{R}\) and \(\varPsi _{x}: I_{2}\mapsto \mathbb{R}\) be defined by
respectively. Then Ψ is said to be coordinate convex on \(I_{1}\times I_{2}\) if \(\varPsi _{y}\) is convex on \(I_{1}\) for all \(y\in I_{2}\) and \(\varPsi _{x}\) is convex on \(I_{2}\) for all \(x\in I_{1}\).
Remark 1.2
Dragomir [86] proved that every convex function is coordinate convex, but not vice versa.
Next, we recall the concept of η-convexity which can be found in the literature [87].
Definition 1.3
(See [87])
Let \(I\subseteq \mathbb{R}\) be an interval, and \(\varPsi : I\mapsto \mathbb{R}\) and \(\eta : \mathbb{R} \times \mathbb{R}\mapsto \mathbb{R}\) be two real-valued functions. Then Ψ is said to be η-convex if the inequality
holds for all \(x, y\in I\) and \(\mu \in [0, 1]\).
Note that the η-convexity reduces to the usual convexity if \(\eta (x, y)=x-y\) in Definition 1.3.
The main purpose of the article is to give a non-trivial example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex but not vice versa, define the coordinate \((\eta _{1}, \eta _{2})\)-convex function and establish its Hermite–Hadamard type inequality.
2 Main results
To begin this section, it is interesting to give the definition of η-convex function defined on rectangle, and give a non-trivial example for a η-convex function defined on rectangle is not convex.
Definition 2.1
Let \(I_{1}, I_{2}\subseteq \mathbb{R}\) be two intervals, and \(\varPsi : I_{1}\times I_{2}\mapsto \mathbb{R}\) and \(\eta : \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}\) be two real-valued functions. Then Ψ is said to be η-convex if the inequality
holds for all \((x, y), (z, w)\in I_{1}\times I_{2}\) and \(\mu \in [0, 1]\).
Example 2.2
Let \(\varPsi : [1,5]\times [1,5]\mapsto \mathbb{R}\) and \(\eta : \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}\) be defined by
Then Ψ is η-convex on \([1,5]\times [1,5]\), but it is not convex.
Proof
Let \(\mu \in [0, 1]\). Then for any \((x, y), (z, w)\in [1,5]\) we have
Note that
It follows from (2.1) and (2.2) that
which shows that Ψ is η-convex on \([1,5]\times [1,5]\). It is easily to verify that Ψ is not convex on \([1,5]\times [1,5]\), for details see [79]. □
Next, we introduce the definition of coordinate \((\eta _{1}, \eta _{2})\)-convexity.
Definition 2.3
Let \(I_{1}, I_{2}\subseteq \mathbb{R}\) be two intervals, \(\varPsi : I_{1}\times I_{2}\mapsto \mathbb{R}\), \(\eta _{1}, \eta _{2}: \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}\) be three real-valued functions, and the partial mappings \(\varPsi _{y}: I_{1} \mapsto \mathbb{R}\) and \(\varPsi _{x}: I_{2}\mapsto \mathbb{R}\) be defined by
Then Ψ is said to be coordinate \((\eta _{1}, \eta _{2})\)-convex on \(I_{1}\times I_{2}\) if \(\varPsi _{y}\) is \(\eta _{1}\)-convex on \(I_{1}\) and \(\varPsi _{x}\) is \(\eta _{2}\)-convex on \(I_{2}\). In particular, if \(\eta _{1}=\eta _{2}=\eta \), then Ψ is said to be coordinate η-convex.
Example 2.4
Let \(\varPsi : [0, \infty )\times [0, \infty ) \mapsto \mathbb{R}\) be defined by \(\varPsi (x, y)=-|x|-y^{2}\), \(\eta _{1}(x, y)=-x-y\) and \(\eta _{2}(x, y)=-x-2y\). Then Ψ is coordinate \((\eta _{1}, \eta _{2})\)-convex on \([0, \infty )\times [0, \infty )\).
Proof
Let \(x_{1}, y_{1}\in [0, \infty )\) and \(\mu \in [0, 1]\). Then for any \((x, y)\in [0, \infty )\) we clearly see that
It follows from (2.3)–(2.6) that
Therefore, Ψ is coordinate \((\eta _{1}, \eta _{2})\)-convex on \([0, \infty )\times [0, \infty )\) follows from (2.7) and (2.8). □
Theorem 2.5
Let \(I_{1}, I_{2}\subseteq \mathbb{R}\)be two interval and \(\eta : \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}\)be a real-valued function. ThenΨis coordinateη-convex on \(I_{1}\times I_{2}\)ifΨisη-convex on \(I_{1}\times I _{2}\).
Proof
Let \((x,y)\in I_{1}\times I_{2}\), \(u, v\in I_{2}\) and \(z, w\in I_{1}\). Then it follows from the η-convexity of the function Ψ on \(I_{1}\times I_{2}\) that
and
Inequalities (2.9) and (2.10) imply that \(\varPsi _{x}\) is η-convex on \(I_{2}\) and \(\varPsi _{y}\) is η-convex on \(I_{1}\). Therefore, Ψ is coordinate η-convex on \(I_{1}\times I_{2}\). □
Example 2.6
Let \(I_{1}=I_{2}=[0, \infty )\), \(\varPsi , \eta : I_{1}\times I_{2}\mapsto [0, \infty )\) be defined by
Then Ψ is coordinate η-convex on \(I_{1}\times I_{2}\) but it is not η-convex on \(I_{1}\times I_{2}\).
Proof
Let \(x, y, u, v, z, w\in [0, \infty )\) and \(\mu \in [0, 1]\). Then it follows from (2.11) that
Inequalities (2.12)–(2.15) imply that
and
Note that
and
Therefore, Ψ is coordinate η-convex on \(I_{1}\times I_{2}\) follows from (2.16)–(2.19).
Next, we prove that Ψ is not η-convex on \(I_{1}\times I _{2}\).
Let \(\mu \in (0, 1)\), \(x=w=1\) and \(y=z=0\). Then (2.11) leads to
From (2.20) and (2.21) we clearly see that Ψ is not η-convex on \(I_{1}\times I_{2}\). □
Next, we establish a Hermite–Hadamard type inequality for the coordinate \((\eta _{1}, \eta _{2})\)-convex function.
Theorem 2.7
Let \(a, b, c, d\in \mathbb{R}\)with \(a< b\)and \(c< d\), \(\varPsi : [a, b]\times [c, d]\mapsto \mathbb{R}\), \(\eta _{1}, \eta _{2}: \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}\)be three real-valued functions such thatΨis coordinate \((\eta _{1}, \eta _{2})\)-convex on \([a, b]\times [c, d]\)and
for all \(x, y\in \mathbb{R}\), where \(M_{\eta _{1}}\)and \(M_{\eta _{2}}\)are two positive constants. Then
Proof
For any fixed \(x\in [a, b]\), \(\varPsi _{x}(y)=\varPsi (x, y)\) is \(\eta _{2}\)-convex on \([c, d]\) due to Ψ is coordinate \((\eta _{1}, \eta _{2})\)-convex on \([a, b]\times [c, d]\). It follows from [77, Theorem 5] that
Integrating each side of inequality (2.23) with respect to the variable x on \([a, b]\) leads to
By similar arguments we have
Adding (2.24) and (2.25) we get the second and third inequalities of (2.22).
Making use of the \((\eta _{1}, \eta _{2})\)-convexity of the function Ψ on \([a, b]\times [c, d]\) and [88, Theorem 5] again we get
Therefore, the first inequality of (2.22) follows from (2.26) and (2.27) with adding \(-\frac{1}{2}M_{\eta _{2}}\) and \(-\frac{1}{2}M_{\eta _{1}}\) respectively, and the last inequality in (2.22) can be derived from (2.28)–(2.31) immediately, with adding \(\frac{1}{4} [M_{\eta _{1}}+M_{\eta _{2}} ]\). □
3 Results and discussion
In the article, we establish a non-trivial example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex and its converse is not true in general. Furthermore, we define a new class of function which is coordinate \((\eta _{1}, \eta _{2})\)-convex function and prove its well-known Hermite–Hadamard type inequality.
4 Conclusion
We find an example for η-convex function defined on rectangle is not convex. The authors define a coordinate \((\eta _{1}, \eta _{2})\)-convex function and prove its results. Our approach may have further applications in the theory of η-convexity.
References
Guessab, A., Schmeisser, G.: Sharp integral inequalities of the Hermite–Hadamard type. J. Approx. Theory 115(2), 260–288 (2002)
Shi, H.-P., Zhang, H.-Q.: Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J. Math. Anal. Appl. 361(2), 411–419 (2010)
Zhou, W.-J., Zhang, L.: Convergence of a regularized factorized quasi-Newton method for nonlinear least squares problems. Comput. Appl. Math. 29(2), 195–214 (2010)
Yang, X.-S., Zhu, Q.-X., Huang, C.-X.: Generalized lag-synchronization of chaotic mix-delayed systems with uncertain parameters and unknown perturbations. Nonlinear Anal., Real World Appl. 12(1), 93–105 (2011)
Zhu, Q.-X., Huang, C.-X., Yang, X.-S.: Exponential stability for stochastic jumping BAM neural networks with time-varying and distributed delays. Nonlinear Anal. Hybrid Syst. 5(1), 52–77 (2011)
Dai, Z.-F., Wen, F.-H.: A modified CG-DESCENT method for unconstrained optimization. J. Comput. Appl. Math. 235(11), 3332–3341 (2011)
Gou, K., Sun, B.: Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions. Appl. Math. Comput. 217(21), 8765–8777 (2011)
Lin, L., Liu, Z.-Y.: An alternating projected gradient algorithm for nonnegative matrix factorization. Appl. Math. Comput. 217(24), 9997–10002 (2011)
Zhang, L., Li, J.-L.: A new globalization technique for nonlinear conjugate gradient methods for nonconvex minimization. Appl. Math. Comput. 217(24), 10295–10304 (2011)
Xiao, C.-E., Liu, J.-B., Liu, Y.-L.: An inverse pollution problem in porous media. Appl. Math. Comput. 218(7), 3649–3653 (2011)
Huang, C.-X., Liu, L.-Z.: Sharp function inequalities and boundness for Toeplitz type operator related to general fractional singular integral operator. Publ. Inst. Math. 92(106), 165–176 (2012)
Zhou, W.-J.: On the convergence of the modified Levenberg–Marquardt method with a nonmonotone second order Armijo type line search. J. Comput. Appl. Math. 239, 152–161 (2013)
Zhang, L., Jian, S.-Y.: Further studies on the Wei–Yao–Liu nonlinear conjugate gradient method. Appl. Math. Comput. 219(14), 7616–7621 (2013)
Li, X.-F., Tang, G.-J., Tang, B.-Q.: Stress field around a strike-slip fault in orthotropic elastic layers via a hypersingular integral equation. Comput. Math. Appl. 66(11), 2317–2326 (2013)
Qin, G.-X., Huang, C.-X., Xie, Y.-Q., Wen, F.-H.: Asymptotic behavior for third-order quasi-linear differential equations. Adv. Differ. Equ. 2013, Article ID 305 (2013)
Zhou, W.-J., Chen, X.-L.: On the convergence of a modified regularized Newton method for convex optimization with singular solutions. J. Comput. Appl. Math. 239, 179–188 (2013)
Wang, M.-K., Chu, Y.-M.: Asymptotical bounds for complete elliptic integrals of the second kind. J. Math. Anal. Appl. 402(1), 119–126 (2013)
Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A power mean inequality involving the complete elliptic integrals. Rocky Mt. J. Math. 44(5), 1661–1667 (2014)
Wang, M.-K., Chu, Y.-M., Jiang, Y.-P.: Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mt. J. Math. 46(2), 679–691 (2016)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl. 2017, Article ID 106 (2017)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Article ID 210 (2017)
Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler–Mascheroni constant. J. Inequal. Appl. 2018, Article ID 118 (2018)
Huang, T.-R., Tan, S.-Y., Ma, X.-Y., Chu, Y.-M.: Monotonicity properties and bounds for the complete p-elliptic integrals. J. Inequal. Appl. 2018, Article ID 239 (2018)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M.: Monotonicity properties and bounds involving the complete elliptic integrals of the first kind. Math. Inequal. Appl. 21(4), 1185–1199 (2018)
Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019)
Qiu, S.-L., Ma, X.-Y., Chu, Y.-M.: Sharp Landen transformation inequalities for hypergeometric functions, with applications. J. Math. Anal. Appl. 474(2), 1306–1337 (2019)
Wang, M.-K., Chu, Y.-M., Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 22(2), 601–617 (2019)
Wang, M.-K., Chu, Y.-M., Zhang, W.: Precise estimates for the solution of Ramanujan’s generalized modular equation. Ramanujan J. 49(3), 653–668 (2019)
Wang, M.-K., Zhang, W., Chu, Y.-M.: Monotonicity, convexity and inequalities involving the generalized elliptic integrals. Acta Math. Sci. 39B(5), 1440–1450 (2019)
Wang, M.-K., Chu, H.-H., Chu, Y.-M.: Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals. J. Math. Anal. Appl. https://doi.org/10.1016/j.jmaa.2019.123388
Huang, C.-X., Yang, Z.-C., Yi, T.-S., Zou, X.-F.: On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differ. Equ. 256(7), 2101–2114 (2014)
Tang, W.-S., Sun, Y.-J.: Construction of Runge–Kutta type methods for solving ordinary differential equations. Appl. Math. Comput. 234, 179–191 (2014)
Huang, C.-X., Guo, S., Liu, L.-Z.: Boundedness on Morrey space for Toeplitz type operator associated to singular integral operator with variable Calderón–Zygmund kernel. J. Math. Inequal. 8(3), 453–464 (2014)
Xie, D.-X., Li, J.: A new analysis of electrostatic free energy minimization and Poisson–Boltzmann equation for protein in ionic solvent. Nonlinear Anal., Real World Appl. 21, 185–196 (2015)
Zhou, W.-J., Wang, F.: A PRP-based residual method for large-scale monotone nonlinear equations. Appl. Math. Comput. 261, 1–7 (2015)
Dai, Z.-F., Chen, X.-H., Wen, F.-H.: A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations. Appl. Math. Comput. 270, 378–386 (2015)
Liu, Y.-C., Wu, J.: Multiple solutions of ordinary differential systems with min-max terms and applications to the fuzzy differential equations. Adv. Differ. Equ. 2015, Article ID 379 (2015)
Fang, X.-P., Deng, Y.-J., Li, J.: Plasmon resonance and heat generation in nanostructures. Math. Methods Appl. Sci. 38(18), 4663–4672 (2015)
Dai, Z.-F.: Comments on a new class of nonlinear conjugate gradient coefficients with global convergence properties. Appl. Math. Comput. 276, 297–300 (2016)
Li, J.-L., Sun, G.-Y., Zhang, R.-M.: The numerical solution of scattering by infinite rough interfaces based on the integral equation method. Comput. Math. Appl. 71(7), 1491–1502 (2016)
Tan, Y.-X., Jing, K.: Existence and global exponential stability of almost periodic solution for delayed competitive neural networks with discontinuous activations. Math. Methods Appl. Sci. 39(11), 2821–2839 (2016)
Duan, L., Huang, C.-X.: Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model. Math. Methods Appl. Sci. 40(3), 814–822 (2017)
Duan, L., Huang, L.-H., Guo, Z.-Y., Fang, X.-W.: Periodic attractor for reaction-diffusion high-order Hopfield neural networks with time-varying delays. Comput. Math. Appl. 73(2), 233–245 (2017)
Wang, W.-S., Chen, Y.-Z.: Fast numerical valuation of options with jump under Merton’s model. J. Comput. Appl. Math. 318, 79–92 (2017)
Huang, C.-X., Liu, L.-Z.: Boundedness of multilinear singular integral operator with a non-smooth kernel and mean oscillation. Quaest. Math. 40(3), 295–312 (2017)
Hu, H.-J., Liu, L.-Z.: Weighted inequalities for a general commutator associated to a singular integral operator satisfying a variant of Hörmander’s condition. Math. Notes 101(5–6), 830–840 (2017)
Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Generalized Lyapunov–Razumikhin method for retarded differential inclusions: applications to discontinuous neural networks. Discrete Contin. Dyn. Syst. 22B(9), 3591–3614 (2017)
Wang, W.-S.: On A-stable one-leg methods for solving nonlinear Volterra functional differential equations. Appl. Math. Comput. 314, 380–390 (2017)
Hu, H.-J., Zou, X.-F.: Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc. Am. Math. Soc. 145(11), 4763–4771 (2017)
Tan, Y.-X., Huang, C.-X., Sun, B., Wang, T.: Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition. J. Math. Anal. Appl. 458(2), 1115–1130 (2018)
Tang, W.-S., Zhang, J.-J.: Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods. Appl. Math. Comput. 323, 204–219 (2018)
Duan, L., Fang, X.-W., Huang, C.-X.: Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting. Math. Methods Appl. Sci. 41(5), 1954–1965 (2018)
Liu, Z.-Y., Wu, N.-C., Qin, X.-R., Zhang, Y.-L.: Trigonometric transform splitting methods for real symmetric Toeplitz systems. Comput. Math. Appl. 75(8), 2782–2794 (2018)
Huang, C.-X., Qiao, Y.-C., Huang, L.-H., Agarwal, R.P.: Dynamical behaviors of a food-chain model with stage structure and time delays. Adv. Differ. Equ. 2018, Article ID 186 (2018)
Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Periodic orbit analysis for the delayed Filippov system. Proc. Am. Math. Soc. 146(11), 4667–4682 (2018)
Wang, J.-F., Chen, X.-Y., Huang, L.-H.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J. Math. Anal. Appl. 469(1), 405–427 (2019)
Wang, J.-F., Huang, C.-X., Huang, L.-H.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear Anal. Hybrid Syst. 33, 162–178 (2019)
Jiang, Y.-J., Xu, X.-J.: A monotone finite volume method for time fractional Fokker–Planck equations. Sci. China Math. 62(4), 783–794 (2019)
Peng, J., Zhang, Y.: Heron triangles with figurate number sides. Acta Math. Hung. 157(2), 478–488 (2019)
Tian, Z.-L., Liu, Y., Zhang, Y., Liu, Z.-Y., Tian, M.-Y.: The general inner-outer iteration method based on regular splittings for the PageRank problem. Appl. Math. Comput. 356, 479–501 (2019)
Wang, W.-S., Chen, Y.-Z., Fang, H.: On the variable two-step IMEX BDF method for parabolic integro-differential equations with nonsmooth initial data arising in finance. SIAM J. Numer. Anal. 57(3), 1289–1317 (2019)
Chu, Y.-M., Wang, M.-K., Qiu, S.-L.: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41–51 (2012)
Zhao, T.-H., Zhou, B.-C., Wang, M.-K., Chu, Y.-M.: On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, Article ID 42 (2019)
Wang, J.-L., Qian, W.-M., He, Z.-Y., Chu, Y.-M.: On approximating the Toader mean by other bivariate means. J. Funct. Spaces 2019, Article ID 6082413 (2019)
Qian, W.-M., Xu, H.-Z., Chu, Y.-M.: Improvements of bounds for the Sándor–Yang means. J. Inequal. Appl. 2019, Article ID 73 (2019)
He, X.-H., Qian, W.-M., Xu, H.-Z., Chu, Y.-M.: Sharp power mean bounds for two Sándor–Yang means. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2627–2638 (2019)
Qian, W.-M., He, Z.-Y., Zhang, H.-W., Chu, Y.-M.: Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean. J. Inequal. Appl. 2019, Article ID 168 (2019)
Chu, Y.-M., Wang, G.-D., Zhang, X.-H.: The Schur multiplicative and harmonic convexities of the complete symmetric function. Math. Nachr. 284(5–6), 653–663 (2011)
Chu, Y.-M., Xia, W.-F., Zhang, X.-H.: The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications. J. Multivar. Anal. 105, 412–421 (2012)
Wu, S.-H., Chu, Y.-M.: Schur m-power convexity of generalized geometric Bonferroni mean involving three parameters. J. Inequal. Appl. 2019, Article ID 57 (2019)
Zhang, X.-M., Chu, Y.-M., Zhang, X.-H.: The Hermite–Hadamard type inequality of GA-convex functions and its applications. J. Inequal. Appl. 2010, Article ID 507560 (2010)
Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Conformable integral inequalities of the Hermite–Hadamard type in terms of GG- and GA-convexities. J. Funct. Spaces 2019, Article ID 6926107 (2019)
Adil Khan, M., Chu, Y.-M., Khan, T.U., Khan, J.: Some new inequalities of Hermite–Hadamard type for s-convex functions with applications. Open Math. 15(1), 1414–1430 (2017)
Adil Khan, M., Hanif, M., Khan, Z.A., Ahmad, K., Chu, Y.-M.: Association of Jensen’s inequality for s-convex function with Csiszár divergence. J. Inequal. Appl. 2019, Article ID 162 (2019)
Khurshid, Y., Adil Khan, M., Chu, Y.-M., Khan, Z.A.: Hermite–Hadamard–Fejér inequalities for conformable fractional integrals via preinvex functions. J. Funct. Spaces 2019, Article ID 3146210 (2019)
Song, Y.-Q., Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: Integral inequalities involving strongly convex functions. J. Funct. Spaces 2018, Article ID 6595921 (2018)
Zaheer Ullah, S., Adil Khan, M., Chu, Y.-M.: Majorization theorems for strongly convex functions. J. Inequal. Appl. 2019, Article ID 58 (2019)
Zaheer Ullah, S., Adil Khan, M., Khan, Z.A., Chu, Y.-M.: Integral majorization type inequalities for the functions in the sense of strong convexity. J. Funct. Spaces 2019, Article ID 9487823 (2019)
Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: The concept of coordinate strongly convex functions and related inequalities. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2235–2251 (2019)
Chu, Y.-M., Adil Khan, M., Ali, T., Dragomir, S.S.: Inequalities for α-fractional differentiable functions. J. Inequal. Appl. 2017, Article ID 93 (2017)
Adil Khan, M., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, Article ID 70 (2018)
Adil Khan, M., Chu, Y.-M., Kashuri, A., Liko, R., Ali, G.: Conformable fractional integrals versions of Hermite–Hadamard inequalities and their generalizations. J. Funct. Spaces 2018, Article ID 6928130 (2018)
Adil Khan, M., Iqbal, A., Suleman, M., Chu, Y.-M.: Hermite–Hadamard type inequalities for fractional integrals via Green’s function. J. Inequal. Appl. 2018, Article ID 161 (2018)
Adil Khan, M., Khurshid, Y., Du, T.-S., Chu, Y.-M.: Generalization of Hermite–Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces 2018, Article ID 5357463 (2018)
Adik Khan, M., Wu, S.-H., Ullah, H., Chu, Y.-M.: Discrete majorization type inequalities for convex functions on rectangles. J. Inequal. Appl. 2019, Article ID 16 (2019)
Dragomir, S.S.: On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwan. J. Math. 5(4), 775–788 (2001)
Delavar, M.R., Dragomir, S.S.: On η-convexity. Math. Inequal. Appl. 20(1), 203–216 (2017)
Eshaghi Gordji, M., Rostamian Delavar, M., Dragomir, S.S.: Some inequalities related to η-convex functions. Available at http://www.ajmaa.org/RGMIA/papers/v18/v18a08.pdf
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This work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485).
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Zaheer Ullah, S., Adil Khan, M. & Chu, YM. A note on generalized convex functions. J Inequal Appl 2019, 291 (2019). https://doi.org/10.1186/s13660-019-2242-0
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DOI: https://doi.org/10.1186/s13660-019-2242-0