On Frame’s inequalities

In this paper, the errors of the two inequalities in Theorem 3.4.20 in the classic “Analytic Inequalities” by Mitrinovic are corrected, and the corresponding inequalities for circular functions and hyperbolic functions are rebuilt.


Introduction
The classic "Analytic Inequalities" by Mitrinovic [1] has been hailed all over the world since it was published in 1970. The influence of this book on the various branches of mathematics cannot be overestimated and will last forever. As the author says in the introduction of his book, "The greater part of the results included have been checked, although this could not, of course, be done for all the results which appear in the book. We hope, however, that there are not many errors, but the very nature of this book is such that it seems impossible to expect it to be entirely free of them. " I find some errors in "Analytic Inequalities" and announce the specific contents.
It is not difficult to find that these two inequalities above on the common interval (0, π/2) are wrong. I read carefully the only one citation [2] by Frame in [1] for Theorem 3.4.20, which was published in the 1944 issue of "The American Mathematical Monthly" in the form of the report of the Mathematical Seminar. We judge that the object of Frame [2] is a right triangle, so the t must be the other two acute angles of a right triangle, that is, t ∈ (0, π/2). We can only find the related contents of (1.1) in [2] is the item "(7)", but the one (1.2) at least did not appear in [2].
By using the analytic method, this paper has come to the corresponding conclusions of (1.1) and (1.2); specifically, these are, in the form of (1.1) and (1.2), the first of two inequalities holds for hyperbolic functions, while the second one must be reconstructed, and reversely for circular functions on the interval (0, π).

Lemma 6
Let B 2n be the even-indexed Bernoulli numbers. Then the series is increasing for n ≥ 1.
Proof Let Then by (2.3) we obtain 1 a n t 2n+2 , Since we know a 0 /b 0 < a 1 /b 1 , and {a n /b n } n≥1 is increasing by Lemma 6. So {a n /b n } n≥0 is increasing, and K(t) = A(t)/B(t) is increasing on (0, π) by Lemma 4. In view of this completes the proof of Lemma 6.
In order to prove (1.6), we need the following lemmas. We introduce a useful auxiliary function H f ,g . For -∞ ≤ a < b ≤ ∞, let f and g be differentiable on (a, b) and g = 0 on (a, b). Then the function H f ,g is defined by The function H f ,g has some good properties and plays an important role in the proof of a monotonicity criterion for the quotient of power series.

Lemma 9 Let
Then the function L(t) has a minimum point t 0 = 2.72078 . . . , and In particular, we see that the double inequality holds for all t ∈ (0, +∞), the constant θ is best possible.
Numerical results show that i(x) > 0 for all x ∈ (0, 0.0040) and i(x) < 0 for all x ∈ (0.0040, π/2). That is, the upper estimate in (5.5) is smaller than the one in (5.2) on the interval (0, 0.0040), meanwhile the upper estimate in (5.2) is smaller than the one in (5.5) on the interval (0.0040, π/2). So these two inequalities (1.4) and (5.2) are not included in each other.
Numerical results show that j(x) > 0 for all x ∈ (0, 0.4878) and i(x) < 0 for all x ∈ (0.4878, π). That is, the upper estimate in (5.5) is smaller than the one in the right hand side of (5.4) on the interval (0, 0.4878), meanwhile the upper estimate in the right hand side of (5.4) is smaller than the one in (5.5) on the interval (0.4878, π). So these two inequalities (1.4) and the right hand side of (5.4) are not included in each other. In a word, inequality (1.4) is not contained in the other improved Cusa-Huygens inequalities showed in [14] and [2] and is stronger than those ones near x = 0.
Remark 3 Using the methods in [15][16][17] and in [18], one can directly prove the inequalities (1.3) and (1.4), (1.5) and (1.6), respectively. A different approach based on the power series expansions, to proving, refinements and generalizations of inequalities of the similar type can be found in [19].

Conclusions
In the present study, we find that there are two wrong inequalities for circular functions in the famous monograph "Analytic Inequalities" by Mitrinovic, and we reestablish two inequalities on this topic and create two corresponding inequalities for hyperbolic functions. These new inequalities are the generalization of the famous Cusa-Huygens inequality, one of them is not contained in other improved Cusa-Huygens inequalities showed in [14] and [2] and is stronger than the ones near x = 0.

Funding
The author's research is supported by the Natural Science Foundation of China grants No. 11471285 and the Natural Science Foundation of China grants No. 61772025.