We do not only give the extensions of the results given by Gill et al. (1997) for logconvex functions but also obtain some new Hadamardtype inequalities for logconvex convex, and convex functions.
1. Introduction
The following inequality is well known in the literature as Hadamard's inequality:
where is a convex function on the interval of real numbers and with This inequality is one of the most useful inequalities in mathematical analysis. For new proofs, note worthy extension, generalizations, and numerous applications on this inequality; see ([1–6]) where further references are given.
Let be on interval in . Then is said to be convex if, for all and ,
(see [5], Page 1). Geometrically, this means that if , and are three distinct points on the graph of with between and , then is on or below chord
Recall that a function is said to be logconvex function if, for all and , one has the inequality (see [5], Page 3)
It is said to be logconcave if the inequality in (1.3) is reversed.
In [7], Toader defined convexity as follows.
Definition 1.1.
The function , is said to be convex, where , if one has
for all and We say that is concave if is convex.
Denote by the class of all convex functions on such that (if ). Obviously, if we choose Definition 1.1 recaptures the concept of standard convex functions on
In [8], Miheşan defined convexity as in the following:
Definition 1.2.
The function , , is said to be convex, where , if one has
for all and .
Denote by the class of all convex functions on for which . It can be easily seen that for convexity reduces to convexity and for , convexity reduces to the concept of usual convexity defined on , .
For recent results and generalizations concerning convex and convex functions, see ([9–12]).
In the literature, the logarithmic mean of the positive real numbers is defined as the following:
(for , we put ).
In [13], Gill et al. established the following results.
Theorem 1.3.
Let be a positive, convex function on . Then
where is a logarithmic mean of the positive real numbers as in (1.6).
For a positive concave function, the inequality is reversed.
Corollary 1.4.
Let be positive convex functions on . Then
If is a positive concave function, then
For some recent results related to the Hadamard's inequalities involving two convex functions, see [14] and the references cited therein. The main purpose of this paper is to establish the general version of inequalities (1.7) and new Hadamardtype inequalities involving two convex functions, two convex functions, or two convex functions using elementary analysis.
2. Main Results
We start with the following theorem.
Theorem 2.1.
Let be convex functions on and with . Then the following inequality holds:
where is a logarithmic mean of positive real numbers.
For a positive concave function, the inequality is reversed.
Proof.
Since are convex functions on , we have
for all and Writing (2.2) for and multiplying the resulting inequalities, it is easy to observe that
for all and
Integrating inequality (2.3) on over , we get
As
the theorem is proved.
Remark 2.2.
By taking and in Theorem 2.1 we obtain (1.7).
Corollary 2.3.
Let be convex functions on and with . Then
If are positive concave functions, then
Proof.
Let be positive convex functions. Then by Theorem 2.1 we have that
for all , whence (2.7). Similarly we can prove (2.8).
Remark 2.4.
By taking and in (2.7) and (2.8) we obtain the inequalities of Corollary 1.4.
We will now point out some new results of the Hadamard type for logconvex, convex, and convex functions, respectively.
Theorem 2.5.
Let be convex functions on and with Then the following inequalities hold:
Proof.
We can write
Using the elementary inequality ( reals) and equality (2.11), we have
Since are convex functions, we obtain
for all and .
Rewriting (2.12) and (2.13), we have
Integrating both sides of (2.14) and (2.15) on over , respectively, we obtain
Combining (2.16), we get the desired inequalities (2.10). The proof is complete.
Theorem 2.6.
Let be convex functions on and with Then the following inequalities hold:
where is a logarithmic mean of positive real numbers.
Proof.
From inequality (2.14), we have
for all and
Using the elementary inequality ( reals) on the right side of the above inequality, we have
Since are convex functions, then we get
Integrating both sides of (2.19) and (2.20) on over , respectively, we obtain
Combining (2.21), we get the required inequalities (2.17). The proof is complete.
Theorem 2.7.
Let be such that is in , where . If is nonincreasing convex function and is nonincreasing convex function on for some fixed then the following inequality holds:
where
Proof.
Since is convex function and is convex function, we have
for all . It is easy to observe that
Using the elementary inequality ( reals), (2.25) on the right side of (2.26) and making the charge of variable and since is nonincreasing, we have
Analogously we obtain
Rewriting (2.27) and (2.28), we get the required inequality in (2.22). The proof is complete.
Theorem 2.8.
Let be such that is in , where . If is nonincreasing convex function and is nonincreasing convex function on for some fixed Then the following inequality holds:
where
Proof.
Since is convex function and is convex function, then we have
for all . It is easy to observe that
Using the elementary inequality ( reals), (2.32) on the right side of (2.33) and making the charge of variable and since is nonincreasing, we have
Analogously we obtain
Rewriting (2.34) and (2.35), we get the required inequality in (2.29). The proof is complete.
Remark 2.9.
In Theorem 2.8, if we choose , we obtain the inequality of Theorem 2.7.
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