New Sharp Bounds for the Modiﬁed Bessel Function of the First Kind and Toader-Qi Mean †

: Let I v ( x ) be he modiﬁed Bessel function of the ﬁrst kind of order v . We prove the double inequality (cid:113) sinh t t cosh 1/ p ( pt ) holds for t > 0 if and only if p ≥ 2/3 and q ≤ ( ln 2 ) / ln π . The corresponding inequalities for means improve already known results.


Introduction
The modified Bessel function of the first kind of order v, denoted by I v (x), is a particular solution of the second-order differential equation ( [1], p. 77) which can be represented explicitly by the infinite series as where Γ (x) is the gamma function [2][3][4]. There are many properties of I v (x), see for example, [5][6][7][8][9][10][11].
In this paper, we are interested in a special case of I v (x), that is, I 0 (x), which is related to Toader-Qi mean of positive numbers a and b defined by TQ (a, b) = 2 π π/2 0 a cos 2 θ b sin 2 θ dθ = √ abI 0 ln a b (see [12][13][14]), where and in what follows a, b > 0 with a = b. It is undoubted that Toader-Qi mean TQ (a, b) is a new newcomer. Recall that some classical means including the arithmetic mean, geometric mean, logarithmic mean, exponential mean and power mean of order p defined by respectively. Clearly, A (a, b) = A 1 (a, b) and G (a, b) = A 0 (a, b). It is known that p → A p (a, b) is increasing on R. A simple relation among these elementary means is the following inequalities: (see [15][16][17][18][19][20][21]). Another interesting relation proven in [22] is that: Let b > a > 0 and t = ln √ a/b. Then those means mentioned above can be represented in terms of hyperbolic functions: Correspondingly, the inequalities mentioned above are equivalent to for t > 0. Let us return to Toader-Qi mean. In 2015, Qi, Shi, Liu and Yang [13] proved that the inequalities hold. Yang and Chu (Theorem 3.3 of [23]) established a series of sharp inequalities for TQ (a, b) and I 0 (t), for example, the inequalities sinh t t 3/4 hold for t > 0 with λ 0 = 0.6766 . . .. Inspired by the inequalities (3) and (4), Yang and Chu conjectured further that the inequality holds, which was proven in Theorem 3.1 of [24] by Yang, Chu and Song. In fact, they proved the following double inequality holds with the best coefficients √ e/π = 0.930 . . . and 1. More inequalities for TQ (a, b) can be seen in [25,26].
Motivated by the inequalities (9) and A 2/3 < I listed in (3), the aim of this paper is to find the best constants p and q such that double inequality holds, or equivalently, for t > 0. Our main results are as follows.

Tools and Lemmas
To prove our results, we need two tools. The first tool was due to Biernacki and Krzyz [27], which play an important role in dealing with the monotonicity of the ratio of power series. Lemma 1 ( [27]). Let A (t) = ∑ ∞ k=0 a k t k and B (t) = ∑ ∞ k=0 b k t k be two real power series converging on (−r, r) (r > 0) with b k > 0 for all k. If the sequence {a k /b k } is increasing (decreasing) for all k, then the function t → A (t) /B (t) is also increasing (decreasing) on (0, r).
The second tool is the so-called "L'Hospital Monotone Rule" (or, for short, LMR), which is very effective in studying the monotonicity of ratios of two functions.
The following two lemmas will be used to prove Proposition 1. Then ∑ ∞ n=0 a n t n converges too for all values of t and in addition lim t→∞ ∑ ∞ n=0 a n t n ∑ ∞ n=0 b n t n = s.

Three Propositions
The proofs of Theorems 1 and 2 rely on the following propositions.

Proof. Expanding in power series yields
By Lemma 3, we see that Direct calculations gives where the last inequality holds due to This shows that the sequence {u n /v n } n≥0 is strictly decreasing, so is where the second limits holds due to Lemma 4, thereby completing the proof.
Using Lemma 2 we can prove the following lemma, which will be use to prove Theorem 2. ∞). Consequently, the double inequality where the weights q and c q = 2 1−1/q if q > 0 and c q = 0 if q < 0 are the best possible. If q ∈ (0, 1/2) ∪ (1, ∞), then the double inequality (17) is reversed.

Remark 3.
The generalized Heronian mean [35] is defined by Let t = ln √ a/b with b > a > 0 and q = w/ (w + 2) > 0. Then Proposition 3 give a best approximation for H w (a, b) by power means: Our proof is clearly concise than Li, Long and Chu's given in [35].

Proofs of Theorem 1 and 2
We are now in a position to prove Theorems 1 and 2.
Proof of Theorem 1. We have As shown in Propositions 1 and 2, the functions F 0 (t) and F 1 (t) are both strictly positive and decreasing on (0, ∞), so is F (t). And, we easily obtain Using the monotonicity of F (t), the desired double inequality follows. This completes the proof.
Proof of Theorem 2. (i) The necessary condition for the right hand side inequality of (12) to hold follows from the limit relation The sufficiency follow from Theorem 1 and the increasing property of p → cosh 1/p (pt) on R.
(ii) The necessary condition for the left hand side inequality of (12) to hold follows from the limit relation lim t→∞ (cosh (qt)) 1/q (sinh t) /t I 0 (t) 2 ≤ 1.
By the increasing property of q → cosh 1/q (qt), to prove the sufficiency, it suffices to prove the left hand side inequality of (12) holds when q = p 0 . From the first inequality of (7) and the second inequality of (18) it follows that for t > 0, which proves the sufficiency, and the proof is completed.

Concluding Remarks
In this paper, we obtained the best constants p and q such that the double inequality (12) holds for t > 0, or equivalently, (11) holds for a, b > 0 with a = b. This improved the result in [24]. We close the paper by giving two remarks on our results.

Remark 4. It was shown in ([20], 5.25) that
Then the double inequality (11) can be extended as

Remark 5.
As a computable bound, the upper bound t −1 sinh t cosh 3/2 (2t/3) for I 0 (t) is superior to those given (6) and (8). In fact, we have and for t > 0. The inequalities (19) are clear, and we have to check (20). Let Differentiation yields we have h 1 (t) < 0 for t > 0, so is h (t). This leads to h (t) < lim t→0 h (t) = 0, which proves the second inequality of (20) holds for t > 0. It is easy to check that However, more problems remain to be researched on this new family of means, for example: (i) checking the monotonicity of this mean with respect to the parameter α; (ii) finding the lower and upper bounds for this mean in terms of elementary means; (iii) comparing this new mean with others.