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

for all with .

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

where

Simple computations lead to

where

We divide the proof into two cases.

Case 1 ().

In this case,

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

Case 2 ().

From (2.5), we have that

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

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

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),

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

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

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

for all with .

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

where

Simple computations lead to

where

We divide the proof into two cases.

Case 1 ().

In this case,

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

Case 2 ().

From (2.28) we have that

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

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

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),

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

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

for .