Estimation type results related to Fejér inequality with applications

This paper deals with some new theorems and inequalities about a Fejér type integral inequality which estimate the difference between the right and middle part in Fejér inequality with new bounds. Also some applications to higher moments of random variables, an error estimate for trapezoidal formula, and some inequalities in connection with special means are given.


Introduction and preliminaries
Throughout the paper, we use an interval I ⊆ R with the nonempty interior I • .
An interesting problem in (1.1) is the estimation of difference for the right-middle part of this inequality which is named in this work as Fejér trapezoidal inequality. In [10], the Fejér trapezoidal inequality related to convex functions has been obtained as follows. g(x) dx dt. (1.2) Also, the following theorem was proved in [11]. It estimates the difference between the right and middle part of (1.1) using Hölder's inequality. Theorem 1.3 Let f : I • ⊆ R → R be a differentiable mapping, a, b ∈ I • with a < b, and w : [a, b] → R + be a differentiable mapping symmetric to a+b 2 . If |f | q is convex on [a, b], q > 1, then the following inequality holds: Motivated by the above-mentioned results, in this work we obtain a new trapezoidal form of Fejér inequality which is different from (1.2) and (1.3). To obtain the main result, we assume that the absolute value of the derivative of the considered function is convex. In what follows, we replace this assumption with the boundedness of the derivative and with a Lipschitzian condition for the derivative of the considered function to obtain new estimation type results. Furthermore, some applications in connection with random variable, trapezoidal formula, and special means are given.
The following lemma holds for symmetric functions as well and will be used to obtain various inequalities in the next sections.
(ii) With the same argument as that used in (i), we can derive (1.5).
Remark 1.5 With the assumptions of Lemma 1.4, if w is a nonnegative function, then we have the following inequalities: The following identity was obtained in [11] and will be used to obtain the main result.

Main results
For the main result, by using Lemma 1.4, Remark 1.5, and Lemma 1.6, we estimate the difference between the right and middle part of (1.1) with a simple and new face without need of using Hölder's inequality in the proof.
Proof From Lemma 1.6, Corollary 1.5, and the convexity of |f |, we have If we change the order of integration in J, then Calculating all inner integrals in J, we get A simple form of J can be obtained as follows: If we use the change of variable On the other hand, since w is symmetric to a+b 2 , we have Remark 2. 2 We can obtain another form of (1.6) in Lemma 1.6. In fact we get where q(t) = p(1t) = -p(t) ≤ 0. Now using (2.2) in the proof of Theorem 2.1 implies another form of (2.1).

Further estimation results
It is known that any convex function defined on the interval [a, b] is bounded and satisfies a Lipschitz condition [13]. So in this section instead of the convexity of derivative we consider the boundedness of the derivative and a Lipschitzian condition for the derivative of the considered function respectively to obtain new estimation type results. Now suppose that the derivative of the considered function is bounded from below and above. Then we can derive an estimation type result related to Fejér inequality.  on (a, b). Assume that f is integrable on [a, b] and there exist constants m < M such that where p(t) is defined in Lemma 1.6.
Proof From Lemma 1.6 we have which implies that Remark 3.2 If in Theorem 3.1 we assume that w is symmetric to a+b 2 , then from Lemma 1.4 we have Also, using Hölder's inequality, we have which implies that Proof If we consider w ≡ 1, then the relations w ∞ = 1 and Estimation for difference between the right and middle terms of (1.1) when the derivative of the function satisfies a Lipschitz condition is our next aim.

Definition 3.4 ([13]) A function f : [a, b] → R is said to satisfy Lipschitz condition on [a, b]
if there is a constant K so that, for any two points x, y ∈ [a, b], Theorem 3.5 Let f : I → R be a mapping that is differentiable on I • , let a, b ∈ I • be points with a < b, and let w : [a, b] → R be a nonnegative integrable mapping that is differentiable on (a, b). Assume that f is integrable on [a, b] and satisfies a Lipschitz condition for some K > 0. Then where p(t) is defined in Lemma 1.6.
Proof From Lemma 1.6 we get Since f satisfies a Lipschitz condition for some K > 0, then Hence Remark 3.6 In Theorem 3.5 assume that w is symmetric to a+b 2 . With the same argument as in Remark 3.2 and using Lemma 1.4, we get Also we have which implies that 2 12 2 12

Random variable
Suppose that for 0 < a < b, w : [a, b] → [0, +∞) is a continuous probability density function related to a continuous random variable X which is symmetric about a+b 2 . Also, for r ∈ R, suppose that the r-moment (1) If we consider f (x) = x r for r ≥ 2 and x ∈ [a, b], then |f (x)| = rx r-1 which is a convex function and so from (2.1) in Theorem 2.1 we have where from the fact that w is symmetric and b a w(x) dx = 1, we have w(x) dx = 1 2 . If r = 1, E(X) is the expectation of the random variable X and from the above inequality, we obtain the following known bound: (4.1) (2) Notice that if w is nonnegative, then Now if we consider f (x) = x r for r ∈ R and x ∈ [a, b], then m = ra r-1 ≤ f (x) = rx r-1 ≤ rb r-1 = M, and so from (3.1) in Theorem 3.1 we have It follows that Therefore If we consider r = 1 in the above inequality, then we recapture (4.1).
(3) If we consider f (x) = x r for r ∈ R and x ∈ [a, b], then the Lipschitz constant K = sup x∈ [a,b] |f (x)| = sup x∈[a,b] rx r-1 is equivalent to So from (3.2) in Theorem 3.5 we have which implies that

Trapezoidal formula
Consider the partition (P) of the interval [a, b] as a = x 0 < x 1 < x 2 < · · · < x n = b. The where Now, if all the conditions of Theorem 2.1 are satisfied for the partition (P) on the interval [a, b], then using inequality (4.2), summing with respect to i from i = 0 to i = n -1, and using the triangle inequality, we obtain So we get the error bound: :

Special means
In the literature, the following means for real numbers a, b ∈ R are well known: A(a, b) = a + b 2 arithmetic mean, L n (a, b) = b n+1a n+1 (n + 1)(ba) 1 n generalized log-mean, n ∈ N, a < b.
Consider f (x) = x n for x > 0, n ∈ N and a differentiable symmetric (to a+b 2 ) mapping w : [a, b] → R + . Theorem 2.1 implies the following inequality: If we consider w ≡ 1 in (4.4), then we recapture the following result.