Abstract
In this paper, we give some inequalities with power exponential functions derived from the left hand side of Becker-Stark’s inequality:
for \(0< x < \pi/2\).
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1 Introduction
Becker-Stark’s inequality is well known:
for \(0< x < \pi/2\). The research of Becker-Stark’s inequality is one of the active areas in mathematical analysis [1–8]. Recently, Zhu [6] gave the following refinement of Becker-Stark’s inequality: For \(0< x<\pi/2\), the inequalities
and
hold, where the constants \(-(\pi^{2} -9)/(6\pi^{4})\) and \(-(10 -\pi^{2} )/\pi ^{4}\) are the best possible. Moreover, from the right hand side of the inequality (1.1), Chen and Cheung [2] gave the following inequality: For \(0< x < \pi/2\), the inequality
holds, where the constants \(\theta= \pi^{2}/12\) and \(\vartheta=1\) are the best possible. In [5], Sun and Zhu gave a simple proof of the results. The above inequality (1.4) is created based on the right hand side of Becker-Stark’s inequality (1.1). However, in this paper we establish some inequalities created based on the left hand side of the inequality (1.1).
2 Results and discussion
Motivated by (1.4), in this paper, we give some inequalities with power exponential functions derived from the left hand side of Becker-Stark’s inequality (1.1). Since we note that \(8/(\pi^{2} -4x^{2})<1\) for \(0< x< (\sqrt{\pi^{2} -8})/2\) and \(8/(\pi^{2} -4x^{2})>1\) for \((\sqrt{\pi^{2} -8})/2 < x <\pi/2\), we obtain the two inequalities as follows.
Theorem 2.1
For \(0< x < (\sqrt{\pi^{2}-8})/2\), we have
with the best possible constant \(\theta= 0\) and the function
Theorem 2.2
For \((\sqrt{\pi^{2}-8})/2 < x < \pi/2\), we have
with the best possible constant \(\theta= 1\) and the function
From Theorems 2.1 and 2.2, we have the best possible constant θ such that
If \(0< x< (\sqrt{\pi^{2} -8})/2\), the constant θ must be \(\theta< 0\) in order to satisfy \(1 \leq\tan{x}/x <(8/(\pi^{2} -4x^{2}))^{\theta}\). On the other hands, if \((\sqrt{\pi^{2} -8})/2 < x <\pi/2\), the constant θ must be \(1 < \theta\) in order to satisfy \(8/(\pi^{2} -4x^{2}) \leq\tan{x}/x < (8/(\pi^{2} -4x^{2}))^{\theta}\). Here, we obtain the two inequalities as follows.
Theorem 2.3
For \(1/2 < x < (\sqrt{\pi^{2}-8})/2\), we have
where the function \(\vartheta(x)\) is in Theorem 2.1.
Corollary 2.4
For \(0 < x < \pi/2\), we do not have the best possible constant ϑ such that
3 Proofs of main theorems
3.1 Proof of Theorem 2.1
Proof of Theorem 2.1
We set
From
for \(0< x < (\sqrt{\pi^{2}-8})/2\), by Bernoulli’s inequality, we have
By the right hand side of the inequality (1.1), for \(0< x < (\sqrt {\pi^{2}-8})/2\),
where
From \(\sqrt{\pi^{2} -8} -2 x >0\) for \(0< x<(\sqrt{\pi^{2} -8})/2\), it suffices to show that
Here, the derivative of \(g(x)\) is
By \(8 -\pi^{2}<0\) and \(\sqrt{\pi^{2} -8} -2 <0\), we have \(g'(x) < 0\) for any \(0< x<(\sqrt{\pi^{2} -8})/2\). Since \(g(x)\) is strictly decreasing for \(0< x<(\sqrt{\pi^{2} -8})/2\), we have
Therefore, we can get
where
Since \(\tan{x}/x\) is strictly increasing for \(0< x<\pi/2\), we have
for any \(0< x<( \sqrt{\pi^{2} -8})/2\). Hence, for \(0< x<( \sqrt{\pi^{2} -8} )/2\), we obtain
where the constant \(\theta=0\). Since \(\vartheta(x)\) is strictly decreasing for \(0< x<(\sqrt{\pi^{2} -8})/2\) and
the constant \(\theta=0\) is the best possible. Therefore, the proof of Theorem 2.1 is complete. □
3.2 Proof of Theorem 2.2
Proof of Theorem 2.2
We set
From
for \((\sqrt{\pi^{2}-8})/2 < x < \pi/2\), by Bernoulli’s inequality, we have
By the inequality (1.3), for \((\sqrt{\pi^{2}-8})/2 < x <\pi/2\),
where
From \((\sqrt{\pi^{2}-8}-\pi ) (\sqrt{\pi^{2}-8}-2 x ) >0\) for \((\sqrt{\pi^{2} -8})/2 < x < \pi/2\), it suffices to show that
We have the derivatives
and
From
and
we have
Since \(h(x)\) is strictly increasing for \((\sqrt{\pi^{2} -8})/2 < x < \pi /2\), we have
Thus, \(g(x)\) is strictly increasing for \((\sqrt{\pi^{2} -8})/2 < x < \pi /2\) and we have
Therefore, we can get
where
Since we have
for any \((\sqrt{\pi^{2} -8})/2 < x<\pi/2\), we obtain
where the constant \(\theta=1\). Since \(\vartheta(x)\) is strictly decreasing for \((\sqrt{\pi^{2} -8})/2 < x< \pi/2\) and
the constant \(\theta=1\) is the best possible. Hence, the proof of Theorem 2.2 is complete. □
3.3 Proof of Theorem 2.3 and Corollary 2.4
We need two lemmas to prove Theorem 2.3.
Lemma 3.1
For \(-1/5 < t < 0\), we have
Proof
We set
then
From \(f'(t) >0\) for \(-1/5 < t < -1/9\) and \(f'(t) <0\) for \(-1/9 < t <0\), \(f(t)\) is strictly increasing for \(-1/5 < t < -1/9\) and \(f(t)\) is strictly decreasing for \(-1/9 < t < 0\). Since
and
we can get \(f(t)>0\) for \(-1/5 < t < 0\). □
Lemma 3.2
For \(0 < s < 1/5\), we have
Proof
We set
then
From \(f'(s) >0\) for \(0 < s < 1/8\) and \(f'(s) <0\) for \(1/8 < s <1/5\), \(f(s)\) is strictly increasing for \(0 < s < 1/8\) and \(f(s)\) is strictly decreasing for \(1/8 < s <1/5\). Since
and
we can get \(f(s)>0\) for \(0 < s < 1/5\). □
Proof of Theorem 2.3
We set
If
then \(-11/100 < t < 0\) for \(1/2 < x < (\sqrt{\pi^{2}-8})/2\), by Lemma 3.1, we can get
If
then \(0 < s < 1/5\) for \(1/2 < x < (\sqrt{\pi^{2} -8})/2\), by Lemma 3.2 and the inequality (1.2), we can get
Since
and
for \(1/2 < x < (\sqrt{\pi^{2} -8})/2\), we obtain
where
It suffices to show that \(g(x) >0\) for \(1/2 < x < (\sqrt{\pi^{2} -8})/2\). We have derivatives
and
Thus, \(h(x)\) is strictly decreasing for \(1/2 < x < (\sqrt{\pi^{2} -8})/2\). From
we have \(g''(x) < 0\) for \(1/2 < x < (\sqrt{\pi^{2} -8})/2\). Therefore, \(g'(x)\) is strictly decreasing for \(x_{1} < x < (\sqrt{\pi^{2} -8})/2\). From
and
there exists uniquely a real number \(x_{1}\) with \(1/2 < x_{1} < (\sqrt{\pi ^{2} -8})/2\) such that \(g'(x_{1}) = 0\). Hence, \(g(x)\) is strictly increasing for \(1/2 < x < x_{1}\) and \(g(x)\) is strictly decreasing for \(x_{1} < x < (\sqrt{\pi^{2} -8})/2\). From
and
we can get \(g(x) >0\) for \(1/2 < x < (\sqrt{\pi^{2} -8})/2\). Hence, the proof of Theorem 2.3 is complete. □
Proof of Corollary 2.4
By Theorem 2.3, for \(1/2 < x < (\sqrt{\pi^{2} -8})/2\), we have the following:
Therefore
The proof of Corollary 2.4 is complete. □
4 Conclusions
In this paper, we gave four inequalities derived from the left hand side of Becker-Stark’s inequality (1.1), which are natural generalizations of the inequality (1.1). Since the value of \(8/(\pi^{2} -4x^{2})\) is less than 1 for \(0< x< (\sqrt {\pi^{2} -8})/2\) and the value of \(8/(\pi^{2} -4x^{2})\) is larger than 1 for \((\sqrt{\pi^{2} -8})/2 < x <\pi/2\), we established the inequalities in Theorems 2.1 and 2.2. By Theorem 2.3, we obtained Corollary 2.4 immediately.
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Acknowledgements
The author would like to thank Professor Mitsuhiro Miyagi and the referees for their helpful suggestions and good advice.
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Nishizawa, Y. Sharp Becker-Stark’s type inequalities with power exponential functions. J Inequal Appl 2015, 402 (2015). https://doi.org/10.1186/s13660-015-0932-9
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DOI: https://doi.org/10.1186/s13660-015-0932-9