Let and be real numbers such that with . Let and be the weighted geometric mean and arithmetic mean, respectively, defined by , and . In particular, consider the above-mentioned means , and . Then the well-known Ky-Fan inequality is

It is well known that Ky-Fan inequality can be obtained from the Levinson inequality [1], see also [2, page 71].

Theorem 1.1.

Let be a real-valued 3-convex function on , then for

In [3], the second author proved the following result.

Theorem 1.2.

Let be a real-valued 3-convex function on and points on , then

In this paper, we will give an improvement and reversion of Ky-Fan inequality as well as some related results.

Lemma 2.1.

Define the function

Then that is, is 3-convex for .

Theorem 2.2.

Define the function

for as in (1.2). Then

(1)for all ,

that is, is log convex in the Jensen sense;

(2) is continuous on , it is also log convex, that is, for ,

with

where , .

Proof.

(1) Let us consider the function

where , are reals.

for . This implies that is 3-convex. Therefore, by (1.2), we have , that is,

This follows that is log convex in the Jensen sense.

(2) Note that is continuous at all points , and since

Since is a continuous and convex in Jensen sense, it is log convex. That is,

which completes the proof.

Corollary 2.3.

For , as in (1.2),

Proof.

Setting , and in Theorem 1.2, we get or

Again setting , and in Theorem 1.2, we get or

Combining both inequalities (2.12), (2.13), we get

Also we have positive for ; therefore, we have

Applying exponentional function, we get

Remark 2.4.

In Corollary 2.3, putting we get an improvement of Ky-Fan inequality.

Theorem 2.5.

Define the function

for as for Theorem 1.1. Then

(1)for all ,

that is, is log convex in the Jensen sense;

(2) is continuous on , it is also log convex. That is for ,

with

where , .

Proof.

The proof is similar to the proof of Theorem 2.2.

Remark 2.6.

Let us note that similar results for difference of power means were recently obtained by Simic in [4].