The Optimal Convex Combination Bounds for Seiffert's Mean

For with , the Seiffert't mean was introduced by Seiffert [1] as follows:

(1.1)

Recently, the inequalities for means have been the subject of intensive research. In particular, many remarkable inequalities for can be found in the literature [2–6]. Seiffert's mean can be rewritten as (see [5, equation (2.4)])

(1.2)

Let , and be the contraharmonic, arithmetic, geometric and harmonic means of two positive real numbers and with . Then

(1.3)

In [7], Seiffert proved that

(1.4)

for all with .

In [8], the authors found the greatest value and the least value such that the double inequality

(1.5)

holds for all with .

For more results, see [9–23].

The purpose of the present paper is to find the greatest values and the least values such that the double inequalities

(1.6)

hold for all with .

Firstly, we present the optimal convex combination bounds of contraharmonic and geometric means for Seiffert's mean as follows.

Theorem 2.1.

The double inequality holds for all with if and only if and .

Proof.

Firstly, we prove that

(2.1)

for all with .

Without loss of generality, we assume that . Let and . Then (1.1) leads to

(2.2)

where

(2.3)

Simple computations lead to

(2.4)

where

(2.5)

We divide the proof into two cases.

Case 1 ( ).

In this case,

(2.6)

Therefore, the second inequality in (2.1) follows from (2.2)–(2.6). Notice that in this case, the second equality in (2.4) becomes

(2.7)

Case 2 ( ).

From (2.5), we have that

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

From (2.17) and (2.18), we clearly see that for ; hence is strictly increasing in , which together with (2.16) implies that there exists such that for and for ; and hence is strictly decreasing in and strictly increasing for . From (2.14) and the monotonicity of , there exists such that for and for ; hence is strictly decreasing in and strictly increasing for . As this goes on, there exists such that is strictly decreasing in and strictly increasing in . Note that if , then the second equality in (2.4) becomes

(2.19)

Thus for all . Therefore, the first inequality in (2.1) follows from (2.2) and (2.3).

Secondly, we prove that is the best possible lower convex combination bound of the contraharmonic and geometric means for Seiffert's mean.

If , then (2.5) (with in place of ) leads to

(2.20)

From this result and the continuity of we clearly see that there exists such that for . Then the last equality in (2.4) implies that for . Thus is decreasing for . Due to (2.4), for , which is equivalent to, by (2.2),

(2.21)

for .

Finally, we prove that is the best possible upper convex combination bound of the contraharmonic and geometric means for Seiffert's mean.

If , then from (1.1) one has

(2.22)

Inequality (2.22) implies that for any there exists such that

(2.23)

for .

Secondly, we present the optimal convex combination bounds of the contraharmonic and harmonic means for Seiffert's mean as follows.

Theorem 2.2.

The double inequality holds for all with if and only if and .

Proof.

Firstly, we prove that

(2.24)

for all with .

Without loss of generality, we assume that . Let and . Then (1.1) leads to

(2.25)

where

(2.26)

Simple computations lead to

(2.27)

where

(2.28)

We divide the proof into two cases.

Case 1 ( ).

In this case,

(2.29)

Therefore, the first inequality in (2.24) follows from (2.25)–(2.29). Notice that in this case, the second equality in (2.27) becomes

(2.30)

Case 2 ( ).

From (2.28) we have that

(2.31)

(2.32)

(2.33)

(2.34)

(2.35)

(2.36)

(2.37)

(2.38)

(2.39)

(2.40)

(2.41)

From (2.40) and (2.41) we clearly see that for ; hence is strictly decreasing in , which together with (2.39) implies that there exists such that for and for , and hence is strictly increasing in and strictly decreasing for . From (2.37) and the monotonicity of , there exists such that for and for ; hence is strictly increasing in and strictly decreasing for . As this goes on, there exists such that is strictly increasing in and strictly decreasing in . Notice that if , then the second equality in (2.27) becomes

(2.42)

Thus for all . Therefore, the second inequality in (2.24) follows from (2.25) and (2.26).

Secondly, we prove that is the best possible upper convex combination bound of the contraharmonic and harmonic means for Seiffert's mean.

If , then (2.28) (with in place of ) leads to

(2.43)

From this result and the continuity of we clearly see that there exists such that for . Then the last equality in (2.27) implies that for . Thus is increasing for . Due to (2.27), for , which is equivalent to, by (2.25),

(2.44)

for .

Finally, we prove that is the best possible lower convex combination bound of the contraharmonic and harmonic means for Seiffert's mean.

If , then from (1.1) one has

(2.45)

Inequality (2.45) implies that for any there exists such that

(2.46)

for .