Inequalities of the Type Hermite–Hadamard–Jensen–Mercer for Strong Convexity

By using the Jensen–Mercer inequality for strongly convex functions, we present Hermite–Hadamard–Mercer inequality for strongly convex functions. Furthermore, we also present some new Hermite-Hadamard-Mercer-type inequalities for diﬀerentiable functions whose derivatives in absolute value are convex.


Introduction
Mathematical inequalities play a vital role in many fields of science. e field of mathematical inequalities and applications has enrolled an exponential improvement in the last two decades with a significant impact in other fields of modern mathematics including engineering [1], mathematical statistics [2], approximation theory [3,4], information theory [5], and other disciplines [6]. In our work, we use Jensen-Mercer inequality for convex and strongly convex functions and also the well-known Hölder's inequality which plays a significant role in functional analysis, mathematical analysis, complex analysis, numerical analysis, statistics and probability, and partial differential equations.
For strongly convex functions, the Jensen-Mercer inequality is proved in [28] as follows.
where n i�1 p i � 1, Substitute n � 2 in (6); we obtain Jensen-Mercer inequality for strongly convex functions as follows: where For more details and recent results related to eorems 3 and 4, see [28][29][30][31]. e following definitions for fractional integrals are important as these definitions will be used throughout this study.

Main Results
In this section, we obtain the Hermite-Hadamard-Mercer inequality for strongly convex functions by using the Jensen-Mercer inequality for strongly convex functions. We derive some new inequalities related to the right and left sides of the Hermite-Hadamard-Mercer type inequalities for differentiable functions whose derivatives in the absolute value are convex.
Proof. By changing the variables a � τx (7), we obtain Multiplying both sides of the above inequality by τ α− 1 and integrating over [0, 1], we get Mathematical Problems in Engineering equivalent to which is the left side of (9).
To obtain the right side of (9), use the definition of strongly convex function ϑ, and we have Multiplying both sides of the above inequality by τ α− 1 and integrating over [0, 1], we have From the above expression, we can write

Mathematical Problems in Engineering
Combining (13) and (17), we obtain (9). □ Theorem 6. Let all the assumptions of eorem 5 hold. en, Proof. Since ϑ is strongly convex, we can write By changing the variables m Multiplying both sides by τ α− 1 , integrating over [0, 1] and in the obtained expression by substituting , we get the left side of (18).
Using the strongly convex function ϑ and the Jensen-Mercer inequality, we obtain Multiplying both sides of the above inequality by τ α− 1 , integrating over [0, 1] and in the obtained expression by Mathematical Problems in Engineering 5 , we get the right side of (18).
□ Corollary 1. If we take α � 1 in eorem 5, then we have Corollary 2. If we take α � 1 in eorem 6, then we have Remark 1. If we take x � m and z � M in (23), we obtain the Hermite-Hadamard inequality for strongly convex function given in [12].

Mathematical Problems in Engineering
For obtaining the right side of (28), use the definition of strongly convex function, and we have By Multiplying both sides of the above inequality by τ α− 1 , integrating over [0, 1] and substituting u � (τ/2)x + ((2 − τ)/2)z and v � ( From the above expression, we obtain the right side of (28).
Proof. We can write that is, Integrating each term, we obtain Mathematical Problems in Engineering Substituting the values of I 1 , I 2 , and I 3 in (36) and simplifying, we obtain Now, multiplying both sides by ((z − x)/2), we get (34). □ Corollary 3. If we take α � 1 in Lemma 1, then we obtain Lemma 2. Let all the assumptions of Lemma 1 hold. en,

Mathematical Problems in Engineering
Proof. We can write Simplifying each term separately, we obtain Using integration by parts and substitution, we get We have Using integration by parts and substitution, we get Similarly, we have Using (43) Proof. Using Lemma 1, we can write (48) Integrating I 1 , I 2 , and I 3 , we have Substituting the values of I 1 , I 2 , and I 3 in (48), we have Mathematical Problems in Engineering and we obtain (47).
Proof. Using Lemma 1 and applying Hölder's inequality, we have Using the convexity of |ϑ ′ | q and the Jensen-Mercer inequality, we get