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
.