In the paper, by directly verifying an inequality which gives a lower bound for the first order modified Bessel function of the first kind, the authors supply a new proof for the complete monotonicity of a difference between the exponential function e ^1/t and the trigamma function 𝜓′( t ) on (0, ∞). In [3, Lemma 2] , the inequality on (0, ∞) was discovered and employed, where 𝜓( t ) denotes the digamma function and Γ is the classical Euler gamma function which may be defined for ℜ( z ) > 0 by The functions 𝜓′( z ) and 𝜓′′( z ) are respectively called the trigamma function and the tetragamma function. As a whole, the derivatives 𝜓 ^(k) ( z ) for k ∈ {0} ∪ are called polygamma functions. An infinitely differentiable function f defined on an interval I is said to be a completely monotonic function on I if it satisfies for all k ∈ {0} ∪ on I . Some properties of the completely monotonic functions can be found in, for example, [2 , 8] . In [5, Theorem 3.1] and [6, Theorem 1.1] , the following theorem was proved by three methods totally. Theorem 1.1. The function is completely monotonic on (0, ∞) and The second main result of the paper [6] is [6, Theorem 1.2] which has been referenced in [4, Section 1.2] and [5, Lemma 2.1] as follows. Theorem 1.2. For k ∈ {0} ∪ and z ≠ 0, let For ℜ( z ) > 0, the function H_k ( z ) has the integral representations and where the hypergeometric series for b_i ∉ {0, −1, −2, ... }, the shifted factorial ( a ) ₀ = 1 and for n > 0 and any real or complex number a, and the modified Bessel function of the first kind for ν ∈ and z ∈ . When k = 0, the integral representations (1.6) and (1.7) may be written as and for ℜ( z ) > 0. Hence, by the well known formula for ℜ( z ) > 0 and n ∈ , see [1, p. 260, 6.4.1] , the function h ( t ) defined by (1.3) has the following integral representation Proposition 1.3 (Hausdorff-Bernstein-Widder Theorem [8, p. 161, Theorem 12b] ). A necessary and sufficient condition that f ( x ) should be completely monotonic for 0 < x < ∞ is that where α ( t ) is non-decreasing and the integral converges for 0 < x < ∞. Combining the complete monotonicity in Theorem 1.1 and the integral representation (1.14) with the necessary and sufficient condition in Proposition 1.3, it was revealed in [6] that Replacing by t in (1.16) yields [6, Theorem 1.3] below. Theorem 1.4. For t > 0, we have We note that the complete monotonicity in Theorem 1.1 is the basis of the inequality (1.17) and some results in the subsequent papers [4 , 5] . The aim of this paper is, with the help of the integral representation (1.14) but without using Proposition 1.3, to supply a new proof of Theorems 1.1 and 1.4 in a converse direction with that in [4 , 5 , 6] . In other words, Theorem 1.4 will be firstly and straightforwardly proved, and then Theorem 1.1 will be done. By the definition of the modified Bessel function I_ν ( z ) in (1.10), it is easy to see that Hence, in order to prove (1.16), it suffices to show which is equivalent to Consequently, the proof of the inequality (1.16), that is, Theorem 1.4, is thus complete. Substituting the inequality (1.16) into the integral representation (1.14) leads to h ( t ) > 0 and for k ∈ on (0, ∞). As a result, the function h ( t ) is completely monotonic on (0, ∞). The limit (1.4) follows immediately from taking t → ∞ on both sides of the integral representation (1.14). Theorem 1.1 is thus proved. Remark 2.1. The inequality (2.1) is equivalent to An immediate differentiation yields Since Q ′′′ ( u ) and Q ′′ (0) = 0, it follows that Q ′′ ( u ) > 0 on (0, ∞). Owing to Q ′(0) = 0 and Q ′′ ( u ) > 0, it is derived that Q ′ ( u ) > 0. Finally, since Q (0) = 0, the function Q ( u ) is positive on (0, ∞). This gives an alternative proof of the inequality (2.1). Remark 2.2. This is a slightly modified version of the preprint [7] .