Reverses of Féjer's Inequalities for Convex Functions

. Let f be a convex function on I and a; b 2 I with a < b: If p : [ a;b ] ! [ a; 1 ) is Lebesgue integrable and symmetric, namely p ( b + a (cid:0) t ) = p ( t ) for all t 2 [ a;b ] ; then we show in this paper that

and

Introduction
The following inequality holds for any convex function f de…ned on R ; a; b 2 R, a < b: It was …rstly discovered by Ch. Hermite in 1881 in the journal Mathesis (see [7]). But this result was nowhere mentioned in the mathematical literature and was not widely known as Hermite's result. E. F. Beckenbach, a leading expert on the history and the theory of convex functions, wrote that this inequality was proven by J. Hadamard in 1893 [1]. In 1974, D. S. Mitrinović found Hermite's note in Mathesis [7]. Since (1.1) was known as Hadamard's inequality, the inequality is now commonly referred as the Hermite-Hadamard inequality. For a monograph devoted to this result see [5]. The recent survey paper [4] provides other related results.
Let f : [a; b] ! R be a convex function on [a; b] and assume that f 0 + (a) and f 0 (b) are …nite. We recall the following improvement and reverse inequality for the …rst Hermite-Hadamard result that has been established in [2] 0 The following inequality that provides a reverse and improvement of the second Hermite-Hadamard result has been obtained in [3] 0 The constant 1 8 is best possible in both (1.2) and (1.3). In 1906, Fejér [6], while studying trigonometric polynomials, obtained inequalities which generalize that of Hermite & Hadamard: where f is a convex function in the interval (a; b) and p is a positive function in the same interval such that i.e., y = p (t) is a symmetric curve with respect to the straight line which contains the point 1 2 (a + b) ; 0 and is normal to the t-axis. Under those conditions the following inequalities are valid: If f is concave on (a; b), then the inequalities reverse in (1.4).
Clearly, for p (t) 1 on [a; b] we get 1.1. If we take p (t) = t a+b 2 ; t 2 [a; b] in Theorem 1, then we have for any convex function f : [a; b] ! R: We observe that, if we take p (t) = (b t) (t a) ; t 2 [a; b] ; then p satis…es the conditions in Theorem 1, and by (1.4) we have the following inequality as well for any convex function f : [a; b] ! R: Motivated by the above results, in this paper we obtain an improvement and a reverse for each inequality in (1.4) and therefore generalize the Hermite-Hadamard inequalities (1.2) and (1.3).

Improvements and Reverse of Féjer Inequalities
Following Roberts and Varberg [8, p. 5 ], we recall that if f : I ! R is a convex function, then for any x 0 2 I (the interior of the interval I) the limits : The functions f 0 and f 0 + are monotonic nondecreasing on I and this property can be extended to the whole interval I (see [8, p. 7 ]).
From the monotonicity of the lateral derivatives f 0 and f 0 + we also have the gradient inequality for any x; y 2 I: ; then at the end points we also have the inequalities for any x 2 (a; b] and for any y 2 [a; b): We have the following re…nement and reverse of Fejer's …rst inequality: Theorem 2. Let f be a convex function on I and a; b 2 I; with a < b: If p : Proof. Let a; b 2 I; with a < b: Using the integration by parts formula for Lebesgue integral, we have By subtracting the second identity from the …rst, we get By the symmetry of p we get and then we can state the following identity of interest in itself By the monotonicity of the derivative we have ; for almost every t 2 a; a + b 2 This implies and by integration If we add these inequalities, then we get Integrating by parts in the Lebesgue integral, we have where for the last equality we used the symmetry of p: Then by (2.3) we obtain the desired result (2.1).

Remark 1.
If we take p 1 in (2.1) and since R b a t a+b 2 = 1 4 (b a) 2 ; hence by (2.1) we recapture the inequalities (1.2) from Introduction.
We also have the following re…nement and reverse of Fejer's second inequality: Proof. Using the integration by parts for Lebesgue integral, we have Observe that By the monotonicity of the derivative we have If we add these inequalities, then we get " If we change the variable s = b + a t; then Finally, observe that and by (2.5) we get (2.4).

Remark 2.
Observe that for p 1 we recapture the inequalities (1.3) from Introduction.
If we consider the symmetric weight p (t) = t a+b 2 ; t 2 [a; b] we obtain from Theorem 2 that and from Theorem 3 that where f is convex on [a; b] : These provide re…nements and reverses of the inequalities (1.5).
If we consider the symmetric weight p (t) = (t a) (b t) ; t 2 [a; b] we obtain from Theorem 2 that and from Theorem 3 that where f is convex on [a; b] : These provide re…nements and reverses of the inequalities (1.6).