Abstract
We provide optimal bounds for the sine and hyperbolic tangent means in terms of various weighted means of the arithmetic and centroidal means
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1 Introduction, definitions and notations
The means
and
defined for positive arguments, have been introduced in [19], 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
Comparing the means and examining the relationships between them is considered important. A cursory overview of MathSciNet shows over 200 papers on this subject, and the number is constantly growing. The aim of this paper is to determine various optimal bounds for the \(\mathsf {M}_{\tanh }\) and \(\mathsf {M}_{\sin }\) with the arithmetic and centroidal means (denoted here by \({\mathsf {A}}\) and \({{\mathsf {C}}}{{\mathsf {e}}}\)). Similar bounds by the arithmetic and contraharmonic means were obtained in [12], and by arithmetic and quadratic means in [11]. For other bounds of Seiffert-like means by the arithmetic and centroidal means, see e.g. [7, 8, 17, 20]. Similar subjects were considered also in [2,3,4,5,6, 10, 13,14,15,16, 18, 21].
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 functions the inequality \(f<g\) holds, then their corresponding means satisfy \(M_f>M_g\). Thus every inequality between means can be replaced by the inequality between their Seiffert functions.
Remark 1
Throughout this paper all means are defined on \((0,\infty )^2\).
Remark 2
Note that the Seiffert function of the centroidal mean \({{\mathsf {C}}}{{\mathsf {e}}}(x,y)=\frac{2}{3}\frac{x^2+xy+y^2}{x+y}\) is \({{\mathsf {c}}}{{\mathsf {e}}}(z)=\frac{3z}{3+z^2}\) 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 \({\mathsf {M}}_{\sin }\) and \({\mathsf {M}}_{\tanh }\) are the functions \(\sin \) and \(\tanh \), respectively.
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 \({\mathsf {A}}<{\mathsf {M}}_{\sin }<{\mathsf {M}}_{\tanh }<{{\mathsf {C}}}{{\mathsf {e}}}\) proven in [19, Lemma 3.1] and Lemma 1.
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
Therefore the problem reduces to finding upper and lower bounds for certain functions defined on the interval (0, 1).
Theorem 1
The inequalities
hold if, and only if, \(\alpha \le \frac{1}{2}\) and \(\beta \ge \frac{3}{\sin 1}-3\approx 0.5652\).
Proof
By formula (2) and Remark 2, we investigate the function
We shall show that h increases. Observe that
Using the known inequalities \(x-x^3/3!<\sin x<x-x^3/3!+x^5/5!\) and \(\cos x<1-x^2/2!+x^4/4!\) we get
so \(h'(z)>0\). We complete the proof by noting that \(\lim _{z \rightarrow 0} h(z)=1/2\).\(\square \)
Theorem 2
The inequalities
hold if, and only if, \(\alpha \le \frac{3}{\tanh 1} -3\approx 0.9391\) and \(\beta \ge 1\).
Proof
We use Remark 2 and formula (2) once more and investigate the function
The function s satisfies \(\lim _{z\rightarrow 0}s(z)=0\) and \(s''(z)=\frac{2}{\sinh ^3 z}\left( \cosh z-\frac{\sinh ^3 z}{z^3}\right) <0\) (by Lemma 2), so s is concave and, by Property 2, its divided difference (and consequently the function h) decreases. To complete the proof note that \(\lim _{z\rightarrow 0} h(z)=1\).\(\square \)
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
The inequalities
hold if, and only if, \(\alpha \le 4\sin 1-3\approx 0.3659\) and \(\beta \ge \frac{1}{2}\).
Proof
According to formula (3), we investigate the function
We shall show that h decreases. We have
The function s satisfies \(s(0)=s'(0)=s''(0)=0\) and
Thus s is negative and so is \(h'\). We complete the proof by noting that \(\lim _{z\rightarrow 0} h(z)=1/2\).\(\square \)
Theorem 4
The inequalities
hold if, and only if, \(\alpha \le 0\) and \(\beta \ge 4\tanh 1-3\approx 0.0464\).
Proof
We use Remark 2 and formula (3) once more and investigate the function
We shall show that h increases. We have
By Lemma 4 we get
Therefore h increases from \(\lim _{z\rightarrow 0} h(z)=0\) to h(1).\(\square \)
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 5
The inequalities
hold if, and only if, \(\alpha \le \frac{1}{2}\) and \(\beta \ge \frac{9}{7}\cot ^2 1\approx 0.5301\).
Proof
Using formula (4) we investigate the function
To show that h increases we use Lemma 3. A simple calculation shows that
and
From \(\sin 2x<2x-(2x)^3/3!+(2x)^5/5!\) and \(\cos 2x>1-(2x)^2/2!+(2x)^4/4!-(2x)^6/6!\) we get
Thus \(r'\) is positive and both r and h increase. We complete the proof by noting that \(\lim \nolimits _{z\rightarrow 0} h(z)=1/2\).\(\square \)
And here comes the hyperbolic tangent version of the previous theorem.
Theorem 6
The inequalities
hold if, and only if, \(\alpha \le \frac{9}{7}(\coth ^2 1-1)\approx 0.9309\) and \(\beta \ge 1\).
Proof
We shall use the identity \(\tanh ^2z=\frac{\cosh 2z-1}{\cosh 2z +1}\).
The function to be considered here is
and its derivative equals
where
and by Lemma 4
This shows that \(h'<0\) so h decreases from \(\lim _{z\rightarrow 0}h(z)=1\) to \(h(1)=\frac{9}{7}(\coth ^2 1-1)\approx 0.9309\).\(\square \)
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 7
The inequalities
hold if, and only if, \(\alpha \le \frac{16\sin ^2 1-9}{7}\approx 0.3327\) and \(\beta \ge \frac{1}{2}\).
Proof
Taking formula (5) into account we should investigate the function
We shall show that h decreases. We have
From \(\sin 2x<2x-(2x)^3/3!+(2x)^5/5!-(2x)^7/7!+(2x)^9/9!\) and \(\cos 2x<1-(2x)^2/2!+(2x)^4/4!-(2x)^6/6!+(2x)^8/8!\) we obtain
Thus \(h'(z)<0\). We complete the proof by noting that \(\lim _{z\rightarrow 0}h(z)=1/2\).\(\square \)
Theorem 8
The inequalities
hold if, and only if, \(\alpha \le 0\) and \(\beta \ge \frac{16\tanh ^2 1-9}{7}\approx 0.0401\).
Proof
This time we investigate the function
This function increases, because by Lemma 4
So the function h assumes values between \(\lim _{z\rightarrow 0} h(z)=0\) and h(1).\(\square \)
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 \({\mathsf {A}}<M<N\), it makes sense to ask what the optimal numbers \(\alpha , \beta \) are satisfying \(N^{\{\alpha \}}< M<N^{\{\beta \}}\). Theorem 6.1 from [19] gives a method for finding such numbers in terms of the Seiffert functions of the means involved. It says
Theorem 9
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 \(N={{\mathsf {C}}}{{\mathsf {e}}}\) we see that \({\widehat{ce}}(z)=\frac{3}{z^2+3}\) and \({\widehat{ce}}^{-1}(x)=\sqrt{3x^{-1}-3}\).
Theorem 10
The inequalities
hold if, and only if, \(\alpha \le \sqrt{\frac{1}{2}}\approx 0.7071\) and \(\beta \ge \sqrt{\frac{3}{\sin 1}-3}\approx 0.7518\).
Proof
Using Theorem 9 we should find the range of the function
The monotonicity of the function \(h^2\) follows from the proof of Theorem 1, so evaluation of the values of h at the endpoints completes the proof.\(\square \)
Theorem 11
The inequalities
hold if, and only if, \(\alpha \le \sqrt{3\coth 1-3}\approx 0.9691 \) and \(\beta \ge 1\).
Proof
According to Theorem 9, we shall consider the function
but we found the range of its square in the proof of Theorem 2.\(\square \)
7 Tools and lemmas
In this section, we place all the technical details needed to prove our main results.
Property 1
A function \(f:(a,b)\rightarrow {\mathbb {R}}\) is convex if, and only if, for every \(a<\theta <b\) its divided difference \(\frac{f(x)-f(\theta )}{x-\theta }\) increases for \(x\ne \theta \).
A simple consequence of Property 1 is
Property 2
If a function \(f:(a,b)\rightarrow {\mathbb {R}}\) is convex and \(\lim _{x\rightarrow a} f(x)=\Theta \), then the function \(\frac{f(x)-\Theta }{x-a}\) increases.
Lemma 1
For all positive \(x\ne y\) the inequality \({\mathsf {M}}_{\tanh }(x,y)<{{\mathsf {C}}}{{\mathsf {e}}}(x,y)\) holds.
Proof
Using Seiffert’s functions we have to proof that \(h(z)=\tanh z-\frac{3z}{3+z^2}>0\) for \(0<z<1\). Note that
This yields
which, combined with \(h(0)=0\) completes the proof.\(\square \)
Lemma 2
(Lazarević [9]) Consider the functions \(g_u:[0,\infty ) \rightarrow {\mathbb {R}}\)
For \(-1/3\le u<0\), the functions \(g_u\) are positive. For \(-1<u<-1/3\), 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 can be found in [1, Theorem 1.25].
Lemma 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 \(\lim _{x\rightarrow b}f(x)=\lim _{x\rightarrow b}g(x)=0\). 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 4
For \(0<x<1\), the following inequalities hold
-
(a)
\(\sinh 3x<3x+\dfrac{(3x)^3}{3!}+\dfrac{(3x)^5}{5!}+\dfrac{(3x)^7}{7!}+2\dfrac{(3x)^9}{9!}\),
-
(b)
\(\sinh 2x<2x+\frac{(2x)^3}{3!}+\frac{(2x)^5}{5!}+2\frac{(2x)^7}{7!}\),
-
(c)
\(\cosh 3x<1+\frac{(3x)^2}{2!}+\frac{(3x)^4}{4!}+\frac{(3x)^6}{6!}+\frac{3}{2}\times \frac{(3x)^8}{8!}\).
Proof
-
a)
$$\begin{aligned} \sinh (3x)&-3x-\dfrac{(3x)^3}{3!}-\dfrac{(3x)^5}{5!}-\frac{(3x)^7}{7!}- \frac{(3x)^9}{9!}\\ {}&=\frac{(3x)^{11}}{11!}+\frac{(3x)^{13}}{13!}+\cdots<\frac{(3x)^9}{9!}\left( \frac{3^2}{10\cdot 11}+\frac{3^4}{ 10\cdot 11\cdot 12\cdot 13}+\cdots \right) \\&\quad <\frac{(3x)^9}{9!}. \end{aligned}$$
Other proofs are similar. \(\square \)
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Nowicka, M., Witkowski, A. Optimal bounds for the sine and hyperbolic tangent means IV. RACSAM 115, 79 (2021). https://doi.org/10.1007/s13398-021-01020-8
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DOI: https://doi.org/10.1007/s13398-021-01020-8