Generalized Lazarevic's Inequality and Its Applications—Part II

Lazarević [1] (or see Mitrinović [2]) gives us the following result.

Theorem 1.1.

Let . Then

(1.1)

holds if and only if .

Recently, the author of this paper gives a new proof of the inequality (1.1) in [3] and extends the inequality (1.1) to the following result in [4].

Theorem 1.2.

Let , and . Then

(1.2)

holds if and only if .

Moreover, the inequality (1.1) can be extended as follows.

Theorem 1.3.

Let or , and . Then

(1.3)

holds if and only if .

Lemma 2.1 (see [5–8]).

Let be two continuous functions which are differentiable on . Further, let on . If is increasing (or decreasing) on , then the functions and are also increasing (or decreasing) on .

Lemma 2.2 (see [9–11]).

Let and be real numbers, and let the power series and be convergent for . If for and if is strictly increasing (or decreasing) for then the function is strictly increasing (or decreasing) on .

Lemma 2.3.

Let and . Then the function strictly increases as increases.

Let , where , and . Then

(3.1)

where , and .

We compute

(3.2)

where

(3.3)

and .

Since

(3.4)

the proof of Theorem 1.3 is complete.

Assuming that and are two different positive numbers, let , , and be the arithmetic, geometric, and logarithmic means, respectively. It is well known that (see [2, 12–16])

(4.1)

Ostle and Terwilliger [17] (or see Leach and Sholander [18], Zhu [16]) gave bounds for in terms of and as follows:

(4.2)

Without loss of generality, let and , then . Replacing with in (1.3), we obtain the following new results for three classical means.

Theorem 4.1.

Let or , and and be two positive numbers such that . Then

(4.3)

holds if and only if .

Now letting in inequality (4.3) be , and , respectively, by Theorem 4.1 and Lemma 2.3 we have the following inequalities:

(4.4)