Convex functions have many applications in various areas of science such as management science, finance and engineering. Moreover, numerous extensions and generalizations of convexity have been applied in mathematical inequalities and optimization. A function \(\psi :J\subseteq \mathbb{R}\to \mathbb{R}\) is called a convex if, for any \(m,n\in J\) and \(\theta \in [a_{1},a_{2}]\),
$$ \psi \bigl(\theta m+(1-\theta )n\bigr)\leq \theta \psi (m)+(1-\theta )\psi (n). $$
The most famous inequality in the literature for convex functions is known as Hadamard’s inequality. This inequality was proposed in 1893 by Hadamard. The following theorem states the double inequality, introduced by Pečarić and Tong ([1], 1992).
Theorem 1
Let
\(\psi :[a_{1},a_{2}]\subseteq \mathbb{R}\to \mathbb{R}\)
be a convex function where
\(a_{1}\leq a_{2}\), then
$$\begin{aligned} \psi \biggl(\frac{a_{1}+a_{2}}{2} \biggr)\leq \frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}}\psi (x)\,dx \leq \frac{\psi (a_{1})+\psi (a_{2})}{2}. \end{aligned}$$
(1)
Hadamard’s inequalities play a crucial role in various branches of science, including engineering, economics, astronomy, and mathematics. Thus, due to its great utility in several areas of pure and applied mathematics, much attention has been paid, by many mathematicians, to Hadamard’s inequality. Consequently, such inequalities were studied extensively by many authors. Also, numerous generalizations and extensions have been reported in a number of papers [2,3,4].
Dragomir [5] defined the following mapping which is considered to be naturally connected with Hadamard’s result:
$$\begin{aligned} H:[0,1]\to \mathbb{R},\qquad H(t):=\frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}} \psi \biggl(tx+(1-t)\frac{a_{1}+a_{2}}{2} \biggr)\,dx, \end{aligned}$$
where \(\psi :[a_{1},a_{2}]\to \mathbb{R}\) is a convex function defined on \([a_{1},a_{2}]\).
Using the above result, we now state the following theorem.
Theorem 2
Let
\(\psi :[a_{1},a_{2}]\to \mathbb{R}\)
be a convex function. We define
\(H:[0,1]\to \mathbb{R}\)
to get the following inequalities:
$$\begin{aligned} \psi \biggl(\frac{a_{1}+a_{2}}{2} \biggr)\leq H(t)\leq \frac{1}{a_{2}-a _{1}} \int _{a_{1}}^{a_{2}}\psi (x)\,dx. \end{aligned}$$
ElFarissi [6] established a new generalization of inequality (1) as follows.
Theorem 3
Assume that
\(\psi :[a_{1},a_{2}]\to \mathbb{R}\)
is a convex function on
\([a_{1},a_{2}]\), then for all
\(\alpha \in [0,1]\)
we have
$$\begin{aligned} \psi \biggl(\frac{a_{1}+a_{2}}{2} \biggr) \leq m(\alpha )\leq \frac{1}{a _{2}-a_{1}} \int _{a_{1}}^{a_{2}}\psi (x)\,dx\leq M(\alpha )\leq \frac{ \psi (a_{1})+\psi (a_{2})}{2}, \end{aligned}$$
(2)
where
$$\begin{aligned} m(\alpha )=:\alpha \psi \biggl(\frac{ta_{2}+(2-t)a_{1}}{2} \biggr)+(1- \alpha )\psi \biggl(\frac{(1+\alpha )a_{2}+(1-\alpha )a_{1}}{2} \biggr) \end{aligned}$$
and
$$\begin{aligned} M(\alpha )=:\frac{1}{2} \bigl[\psi \bigl(\alpha a_{2}+(1-\alpha )a_{1}\bigr)+ \alpha \psi (a_{1})+(1-\alpha )\psi (a_{2}) \bigr]. \end{aligned}$$
The concept of coordinated convexity was introduced by Dragomir [7]. This is a modification of the convex functions, as given by the following definition.
Definition 1
A function \(\psi :A=[a_{1},a_{2}] \times [b_{1},b_{2}] \to \mathbb{R}\) with \(a_{1}< a_{2}\) and \(b_{1}< b_{2}\) is called a convex function on coordinates on A if the partial mappings \(\psi _{x}:[b_{1},b_{2}] \to \mathbb{R}\), \(\psi _{x}(v)=\psi (x,v)\), and \(\psi _{y}:[a_{1},a_{2}] \to \mathbb{R}\), \(\psi _{y}(u)=\psi (u,y)\) defined for all \(x\in [a _{1},a_{2}]\) and \(y\in [b_{1},b_{2}]\), are convex.
Following the above definition, we remark that every convex function, \(\psi :[a_{1},a_{2}]\times [b_{1},b_{2}] \to \mathbb{R}\), is convex on coordinates. However, the converse is not true; see ([7]).
Moreover, a formal definition of the coordinate convex function is given now.
Definition 2
A function \(\psi :A=[a_{1},a_{2}] \times [b_{1},b_{2}] \to \mathbb{R}\) is said to be convex on coordinates on A if the following inequality holds:
$$\begin{aligned} \psi \bigl(sx+(1-s)y,tu+(1-t)v\bigr) \leq& st\psi (x,u)+t(1-s)\psi (y,u)+s(1-t) \psi (x,v)\\ &{}+(1-s) (1-t)\psi (y,v). \end{aligned}$$
The Hadamard inequalities for coordinate convex functions were further modified by Dragomir [7]. These inequalities provide continuous scales of refinements to Hadamard’s inequalities.
Theorem 4
The function
\(\psi :A=[a_{1},a_{2}]\times [b_{1},b_{2}] \to \mathbb{R}\)
is convex on coordinates on
A. Thus the following inequalities hold:
$$\begin{aligned} \psi \biggl(\frac{a_{1}+a_{2}}{2},\frac{b_{1}+b_{2}}{2}\biggr) &\leq \frac{1}{(a_{2}-a_{1})(b_{2}-b_{1})} \int _{a_{1}}^{a_{2}} \int _{b_{1}}^{b_{2}}\psi (x,y)\,dx\,dy \\ & \leq \frac{\psi (a_{1},b_{1})+\psi (a_{1},b_{2})+\psi (a _{2},b_{1})+\psi (a_{2},b_{2})}{4}. \end{aligned}$$
(3)
Many improvements and generalizations of the above result have been extensively investigated by a number of researchers. For several recent results, see [8, 9] and the references therein. While the Hadamard inequalities were proved by considering convexity on coordinates, this paper is aimed at proving the inequality using the definition of convexity. This can be obtained by a change of variables.