A Double Inequality for the Trigamma Function and Its Applications

and Applied Analysis 3 Proof. It follows from the series formulas that ψ 󸀠 (x) = 1 x + 1 2x + 1 6x − 1 30x + ⋅ ⋅ ⋅ , (19) e m/(x+1) = 1 + m x + 1 + 1 2 ( m x + 1 ) 2 + 1 6 ( m x + 1 ) 3 + 1 24 ( m x + 1 ) 5 + ⋅ ⋅ ⋅ , (20) e −m/x = 1 − m x + 1 2 (− m x ) 2 + 1 6 (− m x ) 3 + 1 24 (− m x ) 5 + ⋅ ⋅ ⋅ , (21) and we get lim x→∞ Fm (x) x = lim x→∞ (( 1 (x + 1) + 1 2(x + 1) 2 + 1 6(x + 1) 3 − ((1 + m x + 1 + 1 2 ( m x + 1 ) 2 + 1 6 ( m x + 1 ) 3 − (1 − m x + 1 2 (− m x ) 2 + 1 6 (− m x ) 3 ))


Introduction
For real and positive values of , the classical Euler's gamma function Γ and its logarithmic derivative , the so-called psi function, are defined as For extension of these functions to complex variables and for basic properties, see [1].The derivatives   ,   ,   , . . .are known as polygamma functions (see [2]).In particular,   is called trigamma function.
Our main result is the following Theorem 1.
From Theorem 1, we clearly see the following.

Corollary 2. The double inequality
holds for all  > 0.
Proof.It follows from the series formulas that and we get lim )) (i) If inequality   () ≥ 0 holds for all  > 0, then, from and Lemma 4, we clearly see that (ii) If inequality   () ≤ 0 holds for all  > 0, then and Lemma 4 lead to the conclusion that Lemma 6.Let the function  be defined on (0, ∞) 2 by (9).Then (, 1) and (, ) are not comparable for all  > 0 if  ∈ (1, 2).
The following lemma can be derived immediately from the proof of [4, Theorem 2.1].
The well-known Hermite-Hadamard inequality for convex function can be stated as follows.

Proofs of Theorem 1
Proof of Theorem 1.We divide the proof into four parts.
(I) We prove the first inequality in (7); that is, where   () is defined by (14).It follows from Lemma 10 that where We clearly see that it is enough to prove that () > 0 for  > 0.

Remarks
Remark 14.It follows from (67) and the facts that lim that we clearly see that the upper bound in Theorem 1 for the trigamma function   is better than the upper bounds given in ( 2), ( 3), ( 4), (5), and (6) if  is large enough.
Finally, we give remarks on two mathematical constants  and  (Catalan constant).

Table 1 𝑛
Therefore, (85) provides a new approximation algorithm for .Numerical simulations results carried out with mathematical software show that the given algorithm is more accurate than √6  (see Table