Abstract
We provide optimal bounds for the tangent and hyperbolic sine means in terms of various weighted means of the arithmetic and geometric means.
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1 Introduction, definitions and notation
The means
and
defined for all positive x, y, were introduced in [4], where one of the authors investigates means of the form
It was shown that every symmetric and homogeneous mean of positive arguments can be represented in the form (1) and that every function \(f:(0,1)\rightarrow {\mathbb {R}}\) (called Seiffert function) satisfying
produces a mean. The correspondence between means and Seiffert functions is given by the formula
The aim of this paper is to determine various optimal bounds for the \({{\mathsf {M}}}_{\tan }\) and \(\mathsf {M_{\sinh }}\) with the arithmetic and geometric means (denoted here by \({\mathsf {A}}\) and \({\mathsf {G}}\)).
For two means M, N, the symbol \(M<N\) denotes that for all positive \(x\ne y\) the inequality \(M(x,y)<N(x,y)\) holds.
Our main tool will be the obvious fact that if for two Seiffert means the inequality \(f<g\) holds, then their corresponding means satisfy \(M_f>M_g\). Thus every inequality between means can be expressed in terms of their Seiffert functions.
Remark 1.1
Note that the Seiffert function of the geometric mean \({\mathsf {G}}(x,y)=\sqrt{xy}\) is \({\mathsf {g}}(z)=\frac{z}{\sqrt{1-z^2}}\) and that of the arithmetic mean \({\mathsf {A}}(x,y)=\frac{x+y}{2}\) is the identity function \({\mathsf {a}}(z)=z\). Clearly, the Seiffert functions of \({\mathsf {M}}_{\tan }\) and \({\mathsf {M}}_{\sinh }\) are the functions \(\tan \) and \(\sinh \), respectively.
Remark 1.2
Throughout this paper all means are defined on \((0,\infty )^2\).
For the reader’s convenience in the following sections we place the main results with their proofs, while all lemmas and technical details can be found in the last section of this paper.
The motivation for our research are the inequalities \({\mathsf {G}}<{\mathsf {L}}<{\mathsf {M}}_{\tan }<{\mathsf {M}}_{\sinh }<{\mathsf {A}}\) proven in [4, Lemma 3.2]. The results obtained in this paper show what the distance is between the new and the classical means measured in different ways.
2 Linear bounds
Given three means \(K<L<M\) one may try to find the best \(\alpha ,\beta \) satisfying the double inequality \((1-\alpha )K+\alpha M<L<(1-\beta )K+\beta M\) or equivalently \(\alpha<\frac{L-K}{M-K}<\beta \). If k, l, m are respective Seiffert functions, then the latter can be written as
Thus the problem reduces to finding upper and lower bounds for certain functions defined on the interval (0, 1).
Theorem 2.1
The inequalities
hold if and only if \(\alpha \le \frac{1}{3}\) and \(\beta \ge \cot 1\approx 0.6421\).
Proof
Taking Remark 1.1 and the formula (2) into account we should investigate the function
We shall show that h increases or - which is equivalent - that \(\frac{\frac{z}{\tan z}-1}{\sqrt{1-z^2}-1}\) decreases. By Lemma 7.3 it is enough to prove that the function \(r(z)=\frac{(z/\tan z-1)'}{(\sqrt{1-z^2}-1)'}\) decreases.
We have
and
The function \(s(z)=(8 z^3 - 4 z) \sin z + (-8 z^4 + 8 z^2 - 1) \cos z + \cos 3 z\) satisfies \(s(0)=s'(0)=s''(0)=s'''(0)=0\) and
Thus s is positive and \(r'\) is negative, which shows that h increases. We complete the proof by noting that \(\lim _{z \rightarrow 0} h(z)=1/3\) and \(\lim _{z \rightarrow 1} h(z)=\cot 1.\)\(\square \)
Theorem 2.2
The inequalities
hold if and only if \(\alpha \le \frac{2}{3}\) and \(\beta \ge \frac{1}{\sinh 1}\approx 0.8509 \).
Proof
We use formula (2) once more and investigate the function
We shall show that h increases or - which is equivalent—that \(\frac{\frac{z}{\sinh z}-1}{\sqrt{1-z^2}-1}\) decreases. By Lemma 7.3 it is enough to prove that the function
decreases. A simple calculation reveals that
and
Let \(s(z)=1+3z^2-3 z^4 - (1 -z^2+z^4) \cosh 2z - (z-2 z^3 ) \sinh 2z \). Then
Since for \(n\ge 11\) we have \(z^n<z^5\) and
we can continue Eq. (3)
Since \(s(0)=0\), we conclude that s is positive in (0, 1), so \(r'\) is negative and r decreases and h increases. To complete the proof we note that \(\lim _{z\rightarrow 0} h(z)=2/3\). \(\square \)
3 Harmonic bounds
In this section we look for optimal bounds for means \(K<L<M\) of the form \(\frac{1-\alpha }{M}+\frac{\alpha }{K}<\frac{1}{L}<\frac{1-\beta }{M}+\frac{\beta }{K}\) or, in terms of their Seiffert functions,
We shall use the above to prove two theorems.
Theorem 3.1
The inequalities
hold if and only if \(\alpha \le 0\) and \(\beta \ge \frac{2}{3}\).
Proof
By (4) we shall consider the function
We notice immediately that \(\lim _{z\rightarrow 1} h(z)=0\) and \(\lim _{z\rightarrow 0} h(z)=2/3\) so the only thing we have to show is that 2/3 is the upper bound for h. Note that the inequality \(h(z)<2/3\) is equivalent to \(3\tan z-z-\frac{2z}{\sqrt{1-z^2}}<0\). Substituting \(z=\sin t\) transforms this inequality into \(p(t)=3\tan (\sin t)-\sin t-2\tan t<0\). We have \(p(0)=0\) and
Therefore p is negative in \((0,\pi /2)\), which completes the proof. \(\square \)
And now it is time for the bound of \({\mathsf {M}}_{\sinh }\).
Theorem 3.2
The inequalities
hold if and only if \(\alpha \le 0\) and \(\beta \ge \frac{1}{3}\).
Proof
This time we investigate the function
As in the Proof of Theorem 3.1 we notice that \(\lim _{z\rightarrow 1} h(z)=0\) and \(\lim _{z\rightarrow 0} h(z)=1/3\). We shall show that \(h(z)<1/3\) for all \(0<z<1\). This inequality can be written as \(3\sinh z-2z-\frac{z}{\sqrt{1-z^2}}<0\). Substituting \(z=\sin t\) transforms this inequality into \(p(t)=3\sinh (\sin t)-2\sin t-\tan t<0\). We have \(p(0)=0\) and by Lemma 7.4 we obtain
(the last inequality is valid by the AG inequality). So p is negative and we are done. \(\square \)
4 Quadratic bounds
Given three means \(K<L<M\) one may try to find the best \(\alpha ,\beta \) satisfying the double inequality \(\sqrt{(1-\alpha )K^2+\alpha M^2}<L<\sqrt{(1-\beta )K^2+\beta M^2}\) or equivalently \(\alpha<\frac{L^2-K^2}{M^2-K^2}<\beta \). If k, l, m are respective Seiffert functions, then the latter can be written as
Thus the problem reduces to finding upper and lower bounds for certain functions defined on the interval (0, 1).
Theorem 4.1
The inequalities
hold if and only if \(\alpha \le \frac{1}{3}\) and \(\beta \ge \frac{1}{\tan ^2 1}\approx 0.4123\).
Proof
By formula (5) we should investigate the function
Since \(h'(z)=\frac{2}{\sin ^3 z}\left( \frac{\sin ^3 z}{z^3}-\cos z\right) >0\) (by Lemma 7.1), the function h increases. We complete the proof by noting that \(\lim \limits _{z\rightarrow 0} h(z)=1/3\). \(\square \)
And here comes the hyperbolic sine version of the previous theorem.
Theorem 4.2
The inequalities
hold if and only if \(\alpha \le \frac{2}{3}\) and \(\beta \ge \frac{1}{\sinh ^2 1}\approx 0.7241\).
Proof
The function to be considered here is
Its derivative equals \(h'(z)=\frac{2}{\sinh ^3 z}\left( \frac{\sinh ^3 z}{z^3}-\cosh z\right) .\) By Lemma 7.2 we have that \(h'(z)>0\), so the function h increases. We complete the proof by noting that \(\lim \limits _{z\rightarrow 0} h(z)=2/3\). \(\square \)
5 Bounds with the weighted power mean of order \(-2\)
In this section we look for optimal bounds for means \(K<L<M\) of the form \(\sqrt{\frac{1-\alpha }{M^2}+\frac{\alpha }{K^2}}<\frac{1}{L}<\sqrt{\frac{1-\beta }{M^2}+\frac{\beta }{K^2}}\) or, in terms of their Seiffert functions,
Theorem 5.1
The inequalities
hold if and only if \(\alpha \le 0\) and \(\beta \ge \frac{2}{3}\).
Proof
To prove the theorem we investigate the function
Clearly \(\lim _{z\rightarrow 1}h(z)=0\), which shows that the lower bound for h is \(\alpha =0\). Using Taylor expansion one can easily check that \(\lim _{z\rightarrow 0} h(z)=2/3\). We shall show that this is the best upper bound for h.
Elementary calculations show that the inequality \(h(z)<2/3\) is equivalent to \(3(1-z^2)< (3-z^4)\cos ^2z\). To prove this, note that
\(\square \)
Theorem 5.2
The inequalities
hold if and only if \(\alpha \le 0\) and \(\beta \ge \frac{1}{3}\).
Proof
We follow the same line as in the previous proof. Let
The lower bound of the function h is zero, because \(\lim _{z\rightarrow 1}h(z)=0\). We shall demonstrate that \(\lim _{z\rightarrow 0} h(z)=1/3\) is the best bound for h above.
The inequality \(h(z)<1/3\) is equivalent to \((1-z^2)\cosh ^2z<1-2z^4/3\). Using the Taylor series expansion \(\cosh ^2 z=(\cosh 2z+1)/2=1+z^2+\sum _{n=2}^\infty \frac{2^{2n-1}}{(2n)!}z^{2n}\) we get
which shows that \(h(z)<1/3\). \(\square \)
6 Bounds with varying arguments
If N is a mean, then the formula \(N^{\{t\}}(x,y)=N\left( \frac{x+y}{2}+t\frac{x-y}{2},\frac{x+y}{2}-t\frac{x-y}{2}\right) \) defines a homotopy between the arithmetic mean \({\mathsf {A}}=N^{\{0\}}\) and \(N=N^{\{1\}}\). Therefore if \(N<M<{\mathsf {A}}\) it makes sense to ask what the optimal numbers \(\alpha , \beta \) are satisfying \(N^{\{\alpha \}}< M<N^{\{\beta \}}\). Theorem 6.1 from [4] gives a method for finding such numbers in terms of the Seiffert functions of the means involved. It says
Theorem 6.1
For a Seiffert function k denote \({\widehat{k}}(z)=k(z)/z\). Let M and N be two means with Seiffert functions m and n, respectively. Suppose that \({\widehat{n}}(z)\) is strictly monotone and let \(p_0=\inf \limits _z \frac{{\widehat{n}}^{-1}({\widehat{m}}(z))}{z}\) and \(q_0=\sup \limits _z\frac{{\widehat{n}}^{-1}({\widehat{m}}(z))}{z}\).
If \({\mathsf {A}}(x,y)<M(x,y)<N(x,y)\) for all \(x\ne y\), then the inequalities
hold if and only if \(p\leqslant p_0\) and \(q\geqslant q_0\).
If \(N(x,y)<M(x,y)<{\mathsf {A}}(x,y)\) for all \(x\ne y\), then the inequalities
hold if and only if \(p\leqslant p_0\) and \(q\geqslant q_0\).
In the case of \(N={\mathsf {G}}\) we see that \({\widehat{g}}=\frac{1}{\sqrt{1-z^2}}\) and \({\widehat{g}}^{-1}(x)=\sqrt{1-x^{-2}}\).
Theorem 6.2
The inequalities
hold if and only if \(\alpha \ge \sqrt{\frac{2}{3}} \approx 0.8165\) and \(\beta \le \sqrt{1-\cot ^21}\approx 0.7666\).
Proof
Using Theorem 6.1 we should find the range of the function
A slight modification of the Proof of Theorem 4.1 shows that h decreases from \(\sqrt{2/3} \) to \(\sqrt{1-\cot ^2 1} \), which completes the proof. \(\square \)
Theorem 6.3
The inequalities
hold if and only if \(\alpha \ge \sqrt{\frac{1}{3}}\approx 0.5773 \) and \(\beta \le \sqrt{2-\coth ^2 1}\approx 0.5253\).
Proof
Using Theorem 6.1 we should find the range of the function
We refer to the Proof of Theorem 4.2 to show that h decreases from \(\sqrt{1/3}\) to \(\sqrt{2-\coth ^21} \), which completes the proof. \(\square \)
7 Tools and lemmas
In this section we place all the technical details needed to prove our main results.
Lemma 7.1
(Mitrinović and Adamović [3]) Consider the functions \(f_u:[0,\pi /2) \rightarrow {\mathbb {R}}\)
For \(-1\le u\le -\frac{1}{3}\) the functions \(f_u\) are positive. For \(-\frac{1}{3}<u<0\) there exists \(0<x_u<\frac{\pi }{2}\) such that \(f_u\) is negative in \((0,x_u)\) and positive in \((x_u,\infty )\).
Proof
We have \(f_u(0)=f_u'(0)=0\) and
If \(-1\le u<-1/3\), we have \(\frac{3u+1}{u(u-1)}\le 0\), so \(f_u\) is convex, thus positive.
For \(-1/3<u<0\) the equation \(\tan ^2x-\frac{1+3u}{u(u-1)}=0\) has exactly one solution \(\xi _u\), so \(f_u\) is concave and negative on \((0,\xi _u)\). Then it becomes convex and tends to infinity, thus assumes zero at exactly one point \(x_u\). \(\square \)
Lemma 7.2
(Lazarević [2]) Consider the functions \(g_u:[0,\infty ) \rightarrow {\mathbb {R}}\)
For \(-\frac{1}{3}\le u<0\) the functions \(g_u\) are positive. For \(-1<u<-\frac{1}{3}\) there exists \(x_u>0\) such that \(g_u\) is negative in \((0,x_u)\) and positive in \((x_u,\infty )\).
Proof
We have \(g_u(0)=g_u'(0)=0\) and
If \(-1/3\le u<0\), we have \(\frac{1+3u}{u(u-1)}\ge 0\), so \(g_u\) is convex thus positive. For \(-1<u<-1/3\) the equation \(\tanh ^2x+\frac{1+3u}{u(u-1)}=0\) has exactly one solution \(\xi _u\), so \(g_u\) is concave and negative on \((0,\xi _u)\). Then it becomes convex and tends to infinity, thus assumes zero at exactly one point \(x_u\). \(\square \)
The next lemma comes from [1, Theorem 1.25].
Lemma 7.3
Suppose \(f,g:(a,b)\rightarrow {\mathbb {R}}\) are differentiable with \(g'(x)\ne 0\) and such that \(\lim _{x\rightarrow a}f(x)=\lim _{x\rightarrow a}g(x)=0\) or \(\lim _{x\rightarrow a}f(x)=\lim _{x\rightarrow a}g(x)=0\). Then
-
1.
if \(\frac{f'}{g'}\) is increasing on (a, b), then \(\frac{f}{g}\) is increasing on (a, b),
-
2.
if \(\frac{f'}{g'}\) is decreasing on (a, b), then \(\frac{f}{g}\) is decreasing on (a, b).
Lemma 7.4
For \(0<t<\frac{\pi }{2}\) the inequality \(\cos t\cosh t<1\) holds.
Proof
It follows immediately because
\(\square \)
References
Anderson, G.D., Vamanamurthy, M.K., Vourinen, M.K.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)
Lazarević, I.: Certain inequalities with hyperbolic functions. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 159–170 (1966), 41–48 (in Serbo-Croatian)
Mitrinović, D.S., Adamović, D.D.: Sur une inégalité élémentaire où interviennent des fonctions trigonométriques. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 143–155 (1965), 23–34
Witkowski, A.: On Seiffert-like means. J. Math. Inequal. 4(9), 1071–1092 (2015). https://doi.org/10.7153/jmi-09-83
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Nowicka, M., Witkowski, A. Optimal bounds for the tangent and hyperbolic sine means. Aequat. Math. 94, 817–827 (2020). https://doi.org/10.1007/s00010-020-00705-6
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DOI: https://doi.org/10.1007/s00010-020-00705-6