On some intermediate mean values

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class $\Lambda_{f,g}(a, b)$ of mean values where $f, g$ are continuously differentiable convex functions satisfying the relation $f"(t)=t g"(t), t\in \Bbb R^+$. Then we asked for a characterization of $f, g$ such that the inequalities $H(a, b)\le \Lambda_{f, g}(a, b)\le A(a, b)$ or $L(a, b)\le \Lambda_{f, g}(a, b)\le I(a, b)$ hold for each positive $a, b$, where $H, A, L, I$ are the harmonic, arithmetic, logarithmic and identric means, respectively. For a subclass of $\Lambda$ with $g"(t)=t^s, s\in \Bbb R$, this problem is thoroughly solved.

It is also well known that min{a, b} ≤ H(a, b) ≤ G(a, b) ≤ L(a, b) ≤ I(a, b) ≤ A(a, b) ≤ S(a, b) ≤ max{a, b}, . An inverse problem is to find best possible approximation of a given mean P by elements of an ordered class of means S. A good example for this topic is comparison between the logarithmic mean and the class A s of Hölder means of order s. Namely, since A 0 = lim s→0 A s = G and A 1 = A, it follows from (1) that Since A s is monotone increasing in s, an improving of the above is given by Carlson [2]: Finally, Lin shoved in [3] that is the best possible approximation of the logarithmic mean by the means from the class A s .
Numerous similar results have been obtained recently. For example, an approximation of Seiffert's mean by the class A s is given in [6], [8].
In this article we shall give best possible approximations for a whole variety of elementary means (1) by the class λ s defined below (see Thm 3.).

2.
Let f, g be twice continuously differentiable (strictly) convex functions on R + . By definition (cf [1] if and only if a = b.
It turns out that the expression , represents a mean of two positive numbers a, b; that is, the relation holds for each a, b ∈ R + , if and only if the relation holds for each t ∈ R + .
Let f, g ∈ C ∞ (0, ∞) and denote by Λ the set {(f, g)} of convex functions satisfying the relation (3). There is a natural question how to improve the bounds in (2); in this sense we come upon the following intermediate mean problem: Open question Under what additional conditions on f, g ∈ Λ, the inequalities or, more tightly, hold for each a, b ∈ R + ?
As an illustration, consider the function f s (t) defined to be log t , s = 1, and f ′′ s (t) = t s−2 , s ∈ R, t > 0, it follows that f s (t) is a twice continuously differentiable convex function for s ∈ R, t ∈ R + . Moreover, it is evident that (f s+1 , f s ) ∈ Λ.
We shall give in the sequel a complete answer to the above question concerning the means Those means are obviously symmetric and homogeneous of order one.

As a consequence we obtain some new intermediate mean values; for instance, we show that the inequalities
Note that

Results
We prove firstly the following The expression Λ f,g (a, b) represents a mean of arbitrary numbers a, b ∈ I if and only if the relation holds for t ∈ I.
Remark 1 In the same way, for arbitrary p, q > 0, p + q = 1, it can be deduced that the quotient represents a mean value of numbers a, b if and only if (3) holds.
A generalization of the above assertion is the next Theorem 2 Let f, g : I → R be twice continuously differentiable functions with g ′′ > 0 on I and let p = {p i }, i = 1, 2, · · · , p i = 1 be an arbitrary positive weight sequence. Then the quotient of two Jensen functionals represents a mean of an arbitrary set of real numbers x 1 , x 2 , · · · , x n ∈ I if and only if the relation holds for each t ∈ I.
Remark 2 It should be noted that the relation f ′′ (t) = tg ′′ (t) determines f in terms of g in an easy way. Precisely, where G(t) := t 1 g(u)du and c and d are constants.
Our results concerning the means λ s (a, b), s ∈ R are included in the following Theorem 3 For the class of means λ s (a, b) defined above, the following assertions hold for each a, b ∈ R + .
1. The means λ s (a, b) are monotone increasing in s; The above estimations are best possible.

Proofs
Proof of Theorem 1 We prove firstly the necessity of the condition (3).
From the other hand, due to l'Hospital's rule we obtain Comparing (4) and (5) the desired result follows.
Suppose now that (3) holds and let a < b. Since g ′′ (t) > 0 t ∈ [a, b] by the Cauchy mean value theorem there exists ξ ∈ ( a+t 2 , t) such that But, and, since g ′ is strictly increasing, Therefore, by (6) we get ).
Finally, integrating (7) over t ∈ [a, b] we obtain the assertion from Theorem 1.
Proof of Theorem 2 We shall give a proof of this assertion by induction on n.
By Remark 1, it holds for n = 2.
Next, it is not difficult to check the identity Therefore, by induction hypothesis and Remark 1, we get The inequality min{x 1 , x 2 , · · · , x n } ≤ Λ f,g (p, x), can be proved analogously.
For the proof of necessity, put x 2 = x 3 = · · · = x n and proceed as in Theorem 1.
Remark It is evident from (3) that if I ⊆ R + then f has to be also convex on I. Otherwise, it shouldn't be the case. For example, the conditions of Theorem 2 are satisfied with f (t) = t 3 /3, g(t) = t 2 , t ∈ R. Hence, for an arbitrary sequence {x i } n 1 of real numbers, we obtain Because the above inequality does not depend on n, a probabilistic interpretation of the above result is contained in the following Theorem 4. For an arbitrary probability law F of random variable X with support on (−∞, +∞), we have Proof of Theorem 3, part 1 We shall prove a general assertion of this type. Namely, for an arbitrary positive sequence x = {x i } and an associated weight sequence p = {p i }, i = 1, 2, · · · , denote For s ∈ R, r > 0 we have which is equivalent to This assertion follows applying the result from ( [5], Theorem 2) which states that Lemma 1 For −∞ < a < b < c < +∞, the inequality holds for arbitrary sequences p, x.
A general way to prove the rest of Theorem 3 is to use an easy-checkable identity with t := b−a b+a .
Since 0 < a < b, we get 0 < t < 1. Also, Therefore, we have to compare some one-variable inequalities and to check their validness for each t ∈ (0, 1).
For example, we shall prove that the inequality Since λ s (a, b) is monotone increasing in s, it is enough to prove that By the above formulae, this is equivalent to the assertion that the inequality holds for each t ∈ (0, 1), with We shall prove that the power series expansion of φ(t) have non-positive coefficients. Thus the relation (6) will be proved. Since Hence, c 0 = c 1 = 0; c 2 = −1/90, and, after some calculation, we get c n = 2 (n + 1)(2n + 3) (n + 2) is a negative real number for n ≥ 2. Therefore c n ≤ 0, and the proof of the first part is done.
For 0 < s < 1 we have Therefore, λ s (a, b) > L(a, b) for s > 0 and sufficiently small t : Similarly, we shall prove that the inequality holds for each a, b; 0 < a < b if and only if s ≤ 1.
As before, it is enough to consider the expression It is not difficult to check the identity Hence by (6), we get ψ ′ (t) > 0 i. e. ψ(t) is monotone increasing for t ∈ (0, 1). Therefore By monotonicity it follows that λ s (a, b) ≤ I(a, b) for s ≤ 1.
For s > 1, b−a b+a = t, we have Hence, λ s (a, b) > I(a, b) for s > 1 and t sufficiently small .
From the other hand, Examining the function τ (s), we find out that it has the only real zero at s 0 ≈ 1.0376 and is negative for s ∈ (1, s 0 ).

Remark 2
Since ψ(t) is monotone increasing, we also get A calculation gives 4 log 2 e ≈ 1.0200.
Therefore, applying the assertion from the part 1., we get Finally, we give a detailed proof of the part 7.
Further, we have to show that λ s (a, b) > S(a, b) for some positive a, b whenever s > 5. and the last expression is less than one, it follows that the inequality S(a, b) < λ s (a, b) cannot hold whenever b a is sufficiently large.
The rest of the proof is straightforward.