Abstract
In the article, we present several sharp bounds for the modified Bessel function of the first kind \(I_{0}(t)=\sum_{n=0}^{\infty}\frac{t^{2n}}{2^{2n}(n!)^{2}}\) and the Toader-Qi mean \(TQ(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}a^{\cos^{2}\theta }b^{\sin^{2}\theta}\,d\theta\) for all \(t>0\) and \(a, b>0\) with \(a\neq b\).
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1 Introduction
Let \(a, b>0\), \(p: (0, \infty)\rightarrow\mathbb{R^{+}}\) be a strictly monotone real function, \(\theta\in(0, 2\pi)\) and
Then the mean \(M_{p,n}(a,b)\) was first introduced by Toader in [1] as follows:
where \(p^{-1}\) is the inverse function of p.
From (1.1) and (1.2) we clearly see that
is the classical arithmetic-geometric mean, which is related to the complete elliptic integral of the first kind \({\mathcal{K}}(r)=\int _{0}^{\pi/2} (1-r^{2}\sin^{2}\theta )^{-1/2}\,d\theta\). The Toader mean
is related to the complete elliptic integral of the second kind \({\mathcal{E}}(r)=\int_{0}^{\pi/2} (1-r^{2}\sin^{2}\theta )^{1/2}\,d\theta\). We have
In particular,
is the Toader-Qi mean.
Recently, the arithmetic-geometric mean \(\operatorname{AGM}(a,b)\) and the Toader mean \(T(a,b)\) have attracted the attention of many researchers. In particular, many remarkable inequalities for \(\operatorname{AGM}(a,b)\) and \(T(a,b)\) can be found in the literature [2–20].
For \(q\neq0\), the mean \(M_{x^{q},0}(a, b)\) seems to be mysterious, Toader [1] said that he did not know how to determine any sense for this mean.
Let \(z\in\mathbb{C}\), \(\nu\in\mathbb{R}\backslash\{-1, -2, -3,\ldots\} \) and \(\Gamma(z)=\lim_{n\rightarrow\infty}n!n^{z}/ [\Pi _{k=0}^{\infty}(z+k) ]\) be the classical gamma function. Then the modified Bessel function of the first kind \(I_{\nu}(z)\) [21] is given by
Very recently, Qi et al. [22] proved the identity
and inequalities
for all \(q\neq0\) and \(a, b>0\) with \(a\neq b\), where \(L(a,b)=(b-a)/(\log b-\log a)\), \(A(a,b)=(a+b)/2\), \(G(a,b)=\sqrt{ab}\), and \(I(a,b)=(b^{b}/a^{a})^{1/(b-a)}/e\) are, respectively, the logarithmic, arithmetic, geometric, and identric means of a and b.
Let \(b>a>0\), \(p\in\mathbb{R}\), \(t=(\log b-\log a)/2>0\), and the pth power mean \(A_{p}(a,b)\) be defined by
Then the logarithmic mean \(L(a,b)\), the identric mean \(I(a,b)\), and the pth power mean \(A_{p}(a,b)\) can be expressed as
The main purpose of this paper is to present several sharp bounds for the modified Bessel function of the first kind \(I_{0}(t)\) and the Toader-Qi mean \(TQ(a, b)\).
2 Lemmas
In order to establish our main results we need several lemmas, which we present in this section.
Lemma 2.1
(See [23])
Let \(\binom{n}{k}\) be the number of combinations of n objects taken k at a time, that is,
Then
Lemma 2.2
(See [23])
Let \(\{a_{n}\}_{n=0}^{\infty}\) and \(\{b_{n}\}_{n=0}^{\infty }\) be two real sequences with \(b_{n}>0\) and \(\lim_{n\rightarrow\infty }{a_{n}}/{b_{n}}=s\). Then the power series \(\sum_{n=0}^{\infty }a_{n}t^{n}\) is convergent for all \(t\in\mathbb{R}\) and
if the power series \(\sum_{n=0}^{\infty}b_{n}t^{n}\) is convergent for all \(t\in\mathbb{R}\).
Lemma 2.3
The Wallis ratio
is strictly decreasing with respect to all integers \(n\geq0\) and strictly log-convex with respect to all real numbers \(n\geq0\).
Proof
It follows from (2.1) that
for all integers \(n\geq0\).
Therefore, \(W_{n}\) is strictly decreasing with respect to all integers \(n\geq0\) follows from (2.2).
Let \(f(x)=\Gamma(x+1/2)/\Gamma(x+1)\) and \(\psi(x)=\Gamma^{\prime }(x)/\Gamma(x)\) be the psi function. Then it follows from the monotonicity of \(\psi^{\prime}(x)\) that
for all \(x\geq0\).
Therefore, \(W_{n}\) is strictly log-convex with respect to all real numbers \(n\geq0\) follows from (2.1) and (2.3). □
Lemma 2.4
(See [24])
The double inequality
holds for all \(x>0\) and \(a\in(0, 1)\).
Lemma 2.5
Let \(s_{n}=(2n)!(2n+1)!/[2^{4n}(n!)^{4}]\). Then the sequence \(\{s_{n}\} _{n=0}^{\infty}\) is strictly decreasing and
Proof
The monotonicity of the sequence \(\{s_{n}\}_{n=0}^{\infty}\) follows from
To prove (2.4), we rewrite \(s_{n}\) as
It follows from Lemma 2.4 and (2.5) that
Therefore, equation (2.4) follows from (2.6). □
Lemma 2.6
(See [25])
Let \(A(t)=\sum_{k=0}^{\infty}a_{k}t^{k}\) and \(B(t)=\sum_{k=0}^{\infty}b_{k}t^{k}\) be two real power series converging on \((-r, r)\) (\(r>0\)) with \(b_{k}>0\) for all k. If the non-constant sequence \(\{a_{k}/b_{k}\}\) is increasing (decreasing) for all k, then the function \(A(t)/B(t)\) is strictly increasing (decreasing) on \((0, r)\).
Lemma 2.7
(See [26])
Let \(A(t)=\sum_{k=0}^{\infty}a_{k}t^{k}\) and \(B(t)=\sum_{k=0}^{\infty}b_{k}t^{k}\) be two real power series converging on \(\mathbb{R}\) with \(b_{k}>0\) for all k. If there exists \(m\in\mathbb{N}\) such that the non-constant sequence \(\{a_{k}/b_{k}\}\) is increasing (decreasing) for \(0\leq k\leq m\) and decreasing (increasing) for \(k\geq m\), then there exists \(t_{0}\in(0,\infty)\) such that the function \(A(t)/B(t)\) is strictly increasing (decreasing) on \((0,t_{0})\) and strictly decreasing (increasing) on \((t_{0},\infty)\).
Lemma 2.8
The identity
holds for all \(t\in\mathbb{R}\).
Proof
From (1.4) and Lemma 2.1 together with the Cauchy product we have
□
Lemma 2.9
(See [27])
Let \(-\infty< a< b<\infty\) and \(f, g: [a, b]\rightarrow\mathbb {R}\). Then
if both f and g are increasing or decreasing on \((a, b)\).
Lemma 2.10
(See [28])
Let \(-\infty< a< b<\infty\) and \(f, g: (a, b)\rightarrow\mathbb {R}\). Then
if both f and g are convex on the interval \((a, b)\), and inequality (2.7) becomes an equality if and only if f or g is a linear function on \((a,b)\).
3 Main results
Theorem 3.1
The double inequalities
and
hold for all \(t>0\) and \(b>a>0\).
Proof
From (1.8) we have
and
We clearly see that both \(\cosh(tx)\) and \(1/\sqrt{1-x^{2}}\) are increasing with respect to x on \((0, 1)\). Then Lemma 2.9 and (3.3) lead to
Therefore, inequality (3.1) follows from (3.4) and (3.5).
Let \(t=\log(b/a)/2\). Then it follows from (1.8) and (3.1) that
Therefore, inequality (3.2) follows from (3.6). □
Remark 3.1
From Theorem 3.1 we clearly see that
Theorem 3.2
The double inequalities
and
hold for all \(t>0\) and \(a, b>0\) with \(a\neq b\) if and only if \(\alpha _{1}\leq1/\sqrt{\pi}\), \(\beta_{1}\geq\sqrt{2}/2\), \(\alpha_{2}\leq\sqrt{2/\pi}\) and \(\beta_{2}\geq1\).
Proof
Let
Then simple computation leads to
It follows from Lemma 2.5 and (3.11) that the sequence \(\{a_{n}/b_{n}\} _{n=0}^{\infty}\) is strictly decreasing and
From Lemma 2.8 we have
Lemma 2.6 and (3.13) together with the monotonicity of the sequence \(\{ a_{n}/b_{n}\}_{n=0}^{\infty}\) lead to the conclusion that \(R_{0}(t)\) is strictly decreasing on the interval \((0, \infty)\). Therefore, we have
From Lemma 2.2, (3.12), and (3.13) we know that
Therefore, inequality (3.7) holds for all \(t>0\) if and only if \(\alpha _{1}\leq1/\sqrt{\pi}\) and \(\beta_{1}\geq\sqrt{2}/2\) follows easily from (3.9), (3.14), and (3.15) together with the monotonicity of \(R_{0}(t)\).
Let \(b>a>0\) and \(t=\log(b/a)/2\). Then inequality (3.8) holds for \(a, b>0\) with \(a\neq b\) if and only if \(\alpha_{2}\leq\sqrt{2/\pi}\), and \(\beta_{2}\geq1\) follows from (1.7) and (1.8) together with inequality (3.7) for all \(t>0\) if and only if \(\alpha_{1}\leq1/\sqrt{\pi}\) and \(\beta_{1}\geq\sqrt{2}/2\). □
Remark 3.2
Equations (3.9) and (3.15) imply that
or we have the asymptotic formula
Theorem 3.3
Let \(\lambda_{1}, \lambda_{2}>0\), \(t_{0}=2.7113\ldots\) be the unique solution of the equation
on \((0, \infty)\) and
Then the double inequality
or
holds for all \(t>0\) or \(a,b>0\) with \(a\neq b\) if and only if \(\lambda _{1}\leq2/\pi\), \(\lambda_{2}>\lambda_{0}\).
Proof
Let
and
Then it follows from Lemma 2.2, Lemma 2.5, Lemma 2.8, and (3.19)-(3.21) that
and we have the inequality
for all \(n\geq3\).
From (3.24)-(3.26) we know that the sequence \(\{c_{n}/d_{n}\} _{n=1}^{\infty}\) is strictly increasing for \(1\leq n\leq2\) and strictly decreasing for \(n\geq2\). Then Lemma 2.7 and (3.22) lead to the conclusion that there exists \(t_{0}\in(0, \infty)\) such that \(R_{1}(t)\) is strictly increasing on \((0, t_{0})\) and decreasing on \((t_{0}, \infty)\). Therefore, we have
for all \(t>0\), and \(t_{0}\) is the unique solution of equation (3.16) on \((0, \infty)\).
Note that
From (3.17), (3.19), (3.23), (3.27), and (3.28) we get
Therefore, inequality (3.18) holds for all \(t>0\) if and only if \(\lambda _{1}\leq2/\pi\), \(\lambda_{2}\geq\lambda_{0}\) follows from (3.19) and (3.29) together with the piecewise monotonicity of \(R_{1}(t)\) on \((0, \infty)\). Numerical computations show that \(t_{0}=2.7113\ldots\) and \(\lambda_{0}=0.6766\ldots\) . □
Theorem 3.4
Let \(p, q\in\mathbb{R}\). Then the double inequality
or
holds for all \(t>0\) or \(a, b>0\) with \(a\neq b\) if and only if \(p\geq 3/4\) and \(q\leq3/4\).
Proof
If the first inequality of (3.30) holds for all \(t>0\), then
which implies that \(p\geq3/4\).
It is not difficult to verify that the function \(\cosh^{1-p} t(\sinh t/t)^{p}\) is strictly decreasing with respect to \(p\in\mathbb{R}\) for any fixed \(t>0\), hence we only need to prove the first inequality of (3.30) for all \(t>0\) and \(p=3/4\), that is,
Making use of the power series and Cauchy product formulas together with Lemma 2.8 we have
Let \(W_{n}\) and \(s_{n}\) be, respectively, defined by Lemma 2.3 and Lemma 2.5, and
Then simple computations lead to
It follows from Lemma 2.1 and Lemmas 2.3-2.5 together with (3.33) that
for all \(n\geq4\).
Therefore, inequality (3.31) follows from (3.32)-(3.35).
If the second inequality of (3.30) holds for all \(t>0\), then we have
which implies that \(q\leq3/4\).
Since \(\cosh t>\sinh t/t\), we only need to prove that the second inequality of (3.3) holds for all \(t>0\) and \(q=3/4\), that is,
Let
and \(W_{n}\) be defined by (2.1).
Then simple computations lead to
From (3.38) and (3.39) we clearly see that the sequence \(\{\alpha _{n}/\beta_{n}\}_{n=1}^{\infty}\) is strictly increasing, then Lemma 2.6 and (3.37) lead to the conclusion that the function \((\cosh t-I_{0}(t))/[\cosh t-\sinh t/t]\) is strictly increasing on the interval \((0, \infty)\). Therefore, inequality (3.36) follows from the monotonicity of \((\cosh t-I_{0}(t))/[\cosh t-\sinh t/t]\) and the fact that
□
Theorem 3.5
Let \(p, q>0\), \(t_{0}\) be the unique solution of the equation
and
Then the following statements are true:
-
(i)
The double inequality
$$ 1-\frac{1}{2p^{2}}+\frac{1}{2p^{2}}\cosh(pt)< I_{0}(t)< 1- \frac {1}{2q^{2}}+\frac{1}{2q^{2}}\cosh(qt) $$(3.42)or
$$\begin{aligned}& \biggl(1-\frac{1}{2p^{2}} \biggr)G(a,b)+\frac {1}{2p^{2}}A^{p}_{p}(a,b)G^{1-p}(a,b) \\& \quad < TQ(a,b) < \biggl(1-\frac{1}{2q^{2}} \biggr)G(a,b)+ \frac {1}{2q^{2}}A^{q}_{q}(a,b)G^{1-q}(a,b) \end{aligned}$$holds for all \(t>0\) or \(a, b>0\) with \(a\neq b\) if and only if \(p\leq \sqrt{3}/2\) and \(q\geq1\).
-
(ii)
The inequality
$$ I_{0}(t)\geq1-\frac{\mu_{0}}{p^{2}}+\frac{\mu_{0}}{p^{2}} \cosh(pt) $$(3.43)or
$$ TQ(a,b)\geq \biggl(1-\frac{\mu_{0}}{p^{2}} \biggr)G(a,b)+\frac{\mu _{0}}{p^{2}}A^{p}_{p}(a,b)G^{1-p}(a,b) $$holds for all \(t>0\) or \(a, b>0\) with \(a\neq b\) if \(p\in(\sqrt{3}/2, 1)\).
Proof
(i) Let
Then simple computations lead to
From (3.46) we clearly see that the sequence \(\{u_{n}/v_{n}\} _{n=1}^{\infty}\) is strictly decreasing if \(p\geq1\) and strictly increasing if \(p\leq\sqrt{3}/2\). Then Lemma 2.6 and (3.45) lead to the conclusion that the function \(R_{2}(t)\) is strictly decreasing if \(p\geq1\) and strictly increasing if \(p\leq\sqrt{3}/2\). Hence, we have
for all \(t>0\) if \(p\geq1\) and
for all \(t>0\) if \(p\leq\sqrt{3}/2\).
Therefore, inequality (3.42) holds for all \(t>0\) if \(p\leq\sqrt{3}/2\) and \(q\geq1\) follows easily from (3.44) and (3.47) together with (3.48).
If the first inequality (3.42) holds for all \(t>0\), then we have
which implies that \(p\leq\sqrt{3}/2\).
If there exists \(q_{0}\in(\sqrt{3}/2, 1)\) such that the second inequality of (3.42) holds for all \(t>0\), then we have
But the first inequality of (3.1) leads to
which contradicts inequality (3.49).
(ii) If \(p\in(\sqrt{3}/2, 1)\), then from (3.46) we know that there exists \(n_{0}\in\mathbb{N}\) such that the sequence \(\{u_{n}/v_{n}\} _{n=1}^{\infty}\) is strictly decreasing for \(n\leq n_{0}\) and strictly increasing for \(n\geq n_{0}\). Then (3.45) and Lemma 2.7 lead to the conclusion that there exists \(t_{0}\in(0, \infty)\) such that the function \(R_{2}(t)\) is strictly decreasing on \((0, t_{0}]\) and strictly increasing on \([t_{0}, \infty)\). We clearly see that \(t_{0}\) satisfies equation (3.40). It follows from (3.41) and (3.44) together with the piecewise monotonicity of \(R_{2}(t)\) that
Therefore, inequality (3.43) holds for all \(t>0\) follows from (3.44) and (3.50). □
It is not difficult to verify that the function
is strictly increasing with respect to p on the interval \((0, \infty )\) and
for \(t>0\).
Letting \(p=\sqrt{3}/2, 3/4, \sqrt{2}/2, 2/3, 1/2\) and \(q=1\) in Theorem 3.5(i), then we get Corollary 3.1 immediately.
Corollary 3.1
The inequalities
or
hold for all \(t>0\) or all \(a, b>0\) with \(a\neq b\).
Theorem 3.6
Let \(p>0\). Then the following statements are true:
-
(i)
The inequality
$$ I_{0}(t)>\bigl[\cosh(pt)\bigr]^{\frac{1}{2p^{2}}} $$(3.51)or
$$ TQ(a,b)>G^{1-\frac{1}{2p}}(a,b)A_{p}^{\frac{1}{2p}}(a,b) $$(3.52)holds for all \(t>0\) or \(a, b>0\) with \(a\neq b\) if and only if \(p\geq \sqrt{6}/4\).
-
(ii)
The inequality (3.51) or (3.52) is reversed if and only if \(p\leq1/2\).
-
(iii)
The inequalities
$$ \cosh^{1/2} t< \cosh \biggl(\frac{\sqrt{2}t}{2} \biggr)< \biggl[\cosh \biggl(\frac{\sqrt{6}t}{4} \biggr) \biggr]^{4/3}< I_{0}(t)< \cosh^{2} \biggl(\frac {t}{2} \biggr)< e^{t^{2}/4} $$(3.53)or
$$\begin{aligned} G^{1/2}(a,b)A^{1/2}(a,b) < &A^{\sqrt{2}/2}_{\sqrt{2}/2}(a,b)G^{1-\sqrt {2}/2}(a,b) \\ < &A^{\sqrt{6}/3}_{\sqrt{6}/4}(a,b)G^{1-\sqrt{6}/3}(a,b) \\ < &TQ(a,b)< \frac{A(a,b)+G(a,b)}{2} \\ < &G(a,b)e^{ (A^{2}(a,b)-G^{2}(a,b) )/ (4L^{2}(a,b) )} \end{aligned}$$hold for all \(t>0\) or all \(a, b>0\) with \(a\neq b\).
Proof
(i) If inequality (3.51) holds for all \(t>0\), then we have
which implies that \(p\geq\sqrt{6}/4\).
It follows from Lemma 2 of [29] that the function \([\cosh (pt)]^{1/(2p^{2})}\) is strictly decreasing with respect to \(p\in(0, \infty)\) for any fixed \(t>0\), hence we only need to prove that inequality (3.51) holds for all \(t>0\) and \(p=\sqrt{6}/4\). From the sixth inequality of Corollary 3.1 we clearly see that it suffices to prove that
for all \(t>0\), which is equivalent to
for all \(x>0\), where \(x=\sqrt{6}t/4\).
Let
Then simple computations lead to
From (3.58)-(3.60) and \(\eta_{n}>\eta_{2}>\eta_{1}>0\) for \(n\geq3\) we know that
for all \(x>0\).
Therefore, inequality (3.54) follows easily from (3.55)-(3.57) and (3.61).
(ii) The sufficiency follows easily from the monotonicity of the function \(p\rightarrow [\cosh(pt)]^{1/(2p^{2})}\) and the last inequality in Corollary 3.1 together with the identity \((1+\cosh t)/2=\cosh^{2}(t/2)\).
Next, we prove the necessity. If there exists \(p_{0}\in(1/2, \sqrt {6}/4)\) such that \(I_{0}(t)< [\cosh(p_{0}t)]^{1/(2p_{0}^{2})}\) for all \(t>0\), then we have
But the first inequality of (3.1) leads to
which contradicts (3.62).
(iii) Let \(p=1, \sqrt{2}/2, \sqrt{6}/4, 1/2, 0^{+}\). Then parts (i) and (ii) together with the monotonicity of the function \(p\rightarrow[\cosh(pt)]^{1/(2p^{2})}\) lead to (3.53). □
Theorem 3.7
Let \(\theta\in[0, \pi/2]\). Then the inequality
or
holds for all \(t>0\) or all \(a, b>0\) with \(a\neq b\) if and only if \(\theta\in[\pi/8, 3\pi/8]\). In particular, the inequalities
or
hold for all \(t>0\) or all \(a, b>0\) with \(a\neq b\).
Proof
If inequality (3.63) holds for all \(t>0\), then we have
which implies that \(\theta\in[\pi/8, 3\pi/8]\).
Next, we prove the sufficiency of inequality (3.63). Simple computations lead to
for \(x>0\).
Equation (3.65) and inequality (3.66) imply that the function \(\theta \rightarrow[\cosh(t\cos\theta)+\cosh(t\sin\theta)]\) is decreasing on \([0, \pi/4]\) and increasing on \([\pi/4, \pi/2]\) for any fixed \(t>0\). Hence, it suffices to prove that inequality (3.63) holds for all \(t>0\) and \(\theta=\theta_{0}=\pi/8\).
Let
Then simple computations lead to
for \(n\geq3\).
It follows from Lemma 2.6 and (3.68)-(3.70) that \(R_{3}(t)\) is strictly decreasing on \((0, \infty)\). Therefore,
follows from (3.67) and the monotonicity of \(R_{3}(t)\) together with \(R_{3}(0^{+})=\rho_{0}/\sigma_{0}=1\).
Let \(\theta=\pi/8, \pi/6, \pi/4\). Then inequality (3.64) follows easily from (3.63) and the monotonicity of the function \(\theta\rightarrow [\cosh(t\cos\theta)+\cosh(t\sin\theta)]\). □
Theorem 3.8
The inequality
holds for all \(t>0\).
Proof
It is easy to verify that
for all \(t>0\) and \(x\in(0, 1)\), which implies that the two functions \(1/\sqrt{1-x^{2}}\) and \(\cosh(tx)\) are convex with respect to x on the interval \((0, 1)\). Then from Lemma 2.10 and (3.3) we have
Therefore, inequality (3.72) follows from (3.73). □
Remark 3.3
The inequality \(I_{0}(t)>\sinh(t)/t\) in (3.5) is equivalent to the first inequality \(TQ(a,b)>L(a,b)\) in (1.6). Therefore, Theorem 3.8 is an improvement of the first inequality in (1.6).
Let \(p\in\mathbb{R}\) and \(M(a,b)\) be a bivariate mean of two positive a and b. Then the pth power-type mean \(M_{p}(a,b)\) is defined by
We clearly see that
for all \(\lambda, p\in\mathbb{R}\) and \(a, b>0\) if M is a bivariate mean.
Theorem 3.9
The inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\geq3/4\).
Proof
The second inequality (1.6) can be rewritten as
In [30, 31], the authors proved that the inequality
holds for all distinct positive real numbers a and b with the best possible constant \(2/3\).
Inequalities (3.74) and (3.75) lead to
for all \(a, b>0\) with \(a\neq b\).
If \(p\geq3/4\), then \(TQ(a,b)< I_{3/4}(a,b)\leq I_{p}(a,b)\) follows from (3.76) and the function \(p\rightarrow I_{p}(a,b)\) is strictly increasing [32].
If \(TQ(a,b)< I_{p}(a,b)\) for all \(a, b>0\) with \(a\neq b\). Then
for all \(t>0\).
Inequality (3.77) leads to
which implies that \(p\geq3/4\). □
Remark 3.4
For all \(a, b>0\) with \(a\neq b\), the Toader mean \(T(a,b)\) satisfies the double inequality [5, 7]
with the best possible constants \(3/2\) and \(\log2/(\log\pi-\log2)\), and the one-sided inequality [33]
It follows from (3.78) and (3.79) that
which can be rewritten as
Inequalities (3.74) and (3.80) lead to the inequalities
for all \(a, b>0\) with \(a\neq b\).
Remark 3.5
For all \(a, b>0\) with \(a\neq b\), Theorem 3.4 shows that
It follows from \(L(a,b)< A(a,b)/3+2G(a,b)/3\), given by Carlson in [34], and \(A(a,b)>L(a,b)\) that
Therefore, inequality (3.82) is an improvement of the first and second inequalities of (1.6).
Remark 3.6
In [2, 20, 35], the authors proved that the inequalities
hold for all \(a, b>0\) with \(a\neq b\).
Inequalities (3.81)-(3.83) lead to the chain of inequalities
for all \(a, b>0\) with \(a\neq b\).
Motivated by the first inequality in (3.82) and the third inequality in (3.83), we propose Conjecture 3.1.
Conjecture 3.1
The inequality
holds for all \(a, b>0\) with \(a\neq b\).
For all \(a, b>0\) with \(a\neq b\), inspired by the double inequality
given in Corollary 3.1 and the inequalities
proved by Alzer in [36] we propose Conjecture 3.2.
Conjecture 3.2
The inequality
holds for all \(a, b>0\) with \(a\neq b\).
Remark 3.7
Let \(W_{n}\) be the Wallis ratio defined by (2.1), and \(c_{n}\), \(d_{n}\), and \(s_{n}\) be defined by (3.20). Then it follows from Lemma 2.5 and the proof of Theorem 3.3 that the sequence \(\{s_{n}\}_{n=1}^{\infty}\) is strictly decreasing and \(\lim_{n\rightarrow\infty}s_{n}=2/\pi\), and the sequence \(\{ c_{n}/d_{n}\}_{n=1}^{\infty}\) is strictly increasing for \(n=1, 2\) and strictly decreasing for \(n\geq2\). Hence, we have
and
for all \(n\in\mathbb{N}\).
Inequalities (3.85) and (3.86) lead to the Wallis ratio inequalities
and
for all \(n\in\mathbb{N}\).
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The authors would like to thank the anonymous referee for his/her valuable comments and suggestions. The research was supported by the Natural Science Foundation of China under Grants 11371125, 11401191, and 61374086.
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Yang, ZH., Chu, YM. On approximating the modified Bessel function of the first kind and Toader-Qi mean. J Inequal Appl 2016, 40 (2016). https://doi.org/10.1186/s13660-016-0988-1
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DOI: https://doi.org/10.1186/s13660-016-0988-1