Optimal bounds of classical and non-classical means in terms of Q means

We show optimal bounds of the form Qα<M<Qβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_\alpha<M<Q_\beta $$\end{document}, where Qα(x,y)=A(x,y)A2(x,y)(1-α)A2(x,y)+αG2(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q_\alpha (x,y)={\mathsf {A}}(x,y)\frac{{\mathsf {A}}^2(x,y)}{(1-\alpha ){\mathsf {A}}^2(x,y)+\alpha {\mathsf {G}}^2(x,y)} \end{aligned}$$\end{document}and M belongs to a broad class of classical homogeneous, symmetric means of two variables.

of means. The most popular reference family is the power means given by the formula A r( x, y) = x r +y r 2 1/r and A 0 (x, y) = √ x y.
In this paper we introduce a family of Q-means defined by where x, y > 0, −4 ≤ α ≤ 1 2 , A(x, y) = x+y 2 and G(x, y) = √ x y are the arithmetic and geometric means. It is easy to see that if α < β then Q α < Q β . The Q-means will be used as an alternative reference line for most of the classical means as well as for some recently discovered ones. The reader will find here the optimal bounds for them in terms of Q-means as well as some numerical results. Based on the shapes of the Seiffert means, one of the authors introduced in [20] the notion of Seiffert function: a function f : (0, 1) → R such that is a mean. It was shown that every symmetric and homogeneous mean of positive arguments can be represented in the form (2) and that every function f : (0, 1) → R satisfying produces a mean. The correspondence between a mean and its Seiffert function is given by the formula In [20] it was shown also that the functions sin, tan, sinh and tanh are also Seiffert functions that produce the means called the sine, tangent, hyperbolic sine and hyperbolic tangent means.

Remark 1
Note that due to identity HA = G 2 (where H(x, y) = 2xy x+y is the harmonic mean), (1) can be written as Let us use the formula (4) to calculate the Seiffert function of Q α : In this paper we shall establish the optimal bounds of the form Q α < M < Q β , where M is one of the classical (or less classical) means.

Remark 2
For two means M, N the notation M < N means that M(x, y) < N (x, y) holds unless x = y.

Remark 3
If m, n are the Seiffert functions of M, N respectively, then the inequality M < N is equivalent to n < m by the formula (2).
Note that the inequalities Q α < M < Q β in terms of Seiffert functions can be rewritten as q β (z) < m(z) < q α (z), and, given (5), to find optimal α and β we may use one of the three methods

Remarks on Q means
It might be surprising why we consider the Q α means for −4 ≤ α ≤ 1 2 only. The following lemmas give the answer. (3) can be rewritten as and since inf z∈(0,1) Proof By (4) we have

Main results
At the beginning we consider means that are lesser than the logarithmic mean: -harmonic mean H(x, y) = 2xy x+y = A −1 (x, y). Its Seiffert function equals h(z) = z 1−z 2 , -power mean of order −1/2, The last formula shows that l(z) = artanhz.
All the means mentioned above have one common property: lim x→0 + M(x, 1) = 0, which indicates that there is no lower bound for them in the class of Q means, as the limit of Q α at zero is positive.

Theorem 1 (Bounds for harmonic mean) The inequality
holds if and only if β ≥ −1.
Proof Consider the function from Method 1: It decreases and assumes values in (−∞, −1), which completes the proof. Theorem 2 (Bounds for power mean of order − 1 2 ) The inequality holds if and only if β ≥ − 3 4 . There is no lower bound for the power mean of order −1/2 in the family of Q means.
Proof Consider the function in Method 3 The function g γ satisfies g γ (0) = 0 and (as every even function) g γ (0) = 0. Moreover Both fractions in (6) increase (numerators increase and denominators decrease) thus g γ decreases from −2γ − 3 2 to −∞. This implies that if γ ≥ − 3 4 the function g γ is concave and therefore satisfies g γ (z) < 0 for 0 < z < 1. If γ < − 3 4 the function g γ is convex and thus positive for small z, and cannot preserve sign, because it tends to −∞ at the right end.

Theorem 3 (Bounds for geometric mean)
The inequality G < Q β holds if and only if β ≥ −1/2. There is no Q α that bounds the geometric mean from below.
decreases because both functions in denominator decrease. Therefore it assumes values in −∞, − 1 2 , which completes the proof.
Let us recall that the Seiffert function of the logarithmic mean l(z) = artanhz.

Theorem 4 (Bounds for logarithmic mean). The inequalities
There is no Q α that bounds the logarithmic mean from below.
Proof Let us consider the function from Method 2: Let us investigate the sign of its second derivative.
It is easy to see that if γ ≥ − 1 3 , then u γ is concave and thus negative for all z ∈ (0, 1). For γ < − 1 3 our function is convex and positive for small z and negative for z close to 1, which means that it cannot preserve sign, thus α is not a real number.
There are many interesting means between the logarithmic and the arithmetic means. We shall consider five of them. Here they are: -power mean of order 1 2  The following inequalities between these means can be found in the literature.
Inequality L < A 1/3 has been proven in [11]. This combined with monotonicity of power means shows that L < A 1/2 . Inequalities have been proven in [20] (the means in curly brackets are not comparable).  [12]. Summarizing, we have the following chain of inequalities: Let us begin with the Heronian mean He = 2A+G 3 and the power mean of order 1 2 that can be written as A 1/2 = A+G 2 . Both are members of the family of means that interpolate between the geometric and the arithmetic means given by He p = (1 − p)G + pA, 0 ≤ p ≤ 1. Obviously He = He 2/3 . The Seiffert function for He p is which shows that this function decreases with z ∈ (0, 1) and assumes values in p−1 p , p−1 2 .
For p < 1 5 , we have ( p − 1)/ p < −4, so the corresponding Q is not a mean by Lemma 1.

Corollary 1
The optimal inequalities hold:

Corollary 2
The optimal inequalities hold: And now it's time for three other means It satisfies g γ (0) = g γ (0) = 0 and g γ (1) = 1 − γ − tan 1. Moreover, The function sin z cos 3 z = 1 cos z · 1 cos z · tan z is a product of three positive, increasing and convex functions, so it is convex. By Property 2 its divided difference sin z z cos 3 z increases from m = 1 to M = sin 1 cos 3 1 . Thus we see that if 3γ + 1 ≥ 0, then g γ is concave and therefore negative. On the other hand, if 3γ + 1 < 0 then g γ is convex, and thus positive for small values of z and changes its convexity at most once. Therefore it remains nonnegative in the unit interval if and only if g γ (1) = 1 − γ − tan 1 ≥ 0.
The function sinh z z increases by Property 2 from 1 to infinity, so if 6γ + 1 ≥ 0 the function g γ is concave, and thus negative, while if 6γ + 1 < 0 the function g γ is convex and positive for small z and then remains convex or becomes concave. As a consequence it remains positive on (0, 1) if and only if g γ (1) ≥ 0.
Next we shall take care of several means that are greater than the arithmetic mean: -Neuman-Sándor mean called also inverse hyperbolic sine mean NS(x, y) = |x−y| . Its Seffert mean is ce(z) = 3z 3+z 2 , -root-mean square or quadratic mean RMS(x, y) = with Seiffert function rms(z) = z √ 1+z 2 , -contraharmonic mean C(x, y) = x 2 +y 2 x+y where c(z) = z 1+z 2 . We have The first four inequalities come from [20], next from [13], the remaining two can be found in many sources.
In this part of our paper Method 1 proves to be very useful.
The function tanh is concave and lim z→0 + tanh z = 0, so by Property 2 the function tanh z z decreases and this implies that the function h used in Method 1 decreases. We complete the proof by noting lim z→0 + h(z) =  Proof Once more we use Method 1. The function decreases, so it assumes values between h(1) = 1 and h(0) = 1 2 . Tables 1 on page 9 and Table 2 on page 10 represent absolute and relative errors of the bounds calculated in the preceding section.

Remarks about other one-parameter families of means
In this section we consider two more families of one-parameter means given by formulae similar to (1) and (1 )

41.42
All the maxima in the rightmost column occur at 0 + where RMS(x, y) = x 2 +y 2 2 is the quadratic (or root-mean square) mean and C(x, y) = x 2 +y 2 x+y is the contraharmonic mean. Using (4) we see that the corresponding Seiffert functions are of the form z + αz 3 , so by Lemma 1 V α are means if and only if − 1 2 ≤ α ≤ 4 and V α = Q −α . The second family is where Ce(x, y) = 2 3 x 2 +xy+y 2 x+y is the centroidal mean. We use (4) to find out that their Seiffert functions are of the form w α (z) = z + α 3 z 3 . Again by Lemma 1 we see that W α are means if and only if − 3 2 ≤ α ≤ 12 and W α = Q α/3 .

Tools and lemmas
In this section, we place all the technical details needed to prove our main results. : (a, b) → R is convex if, and only if, for every a < θ < b its divided difference f (x)− f (θ )

Property 1 A function f
x−θ increases for x = θ .
A simple consequence of Property 1 is  on (a, b), then f g is increasing on (a, b), 2. if f g is decreasing on (a, b), then f g is decreasing on (a, b).