Pick functions related to the triple Gamma function

Let Γ3 be the triple gamma function. Our main objective is to prove that the functionlog⁡Γ3(z+1)−Γ3′(1)zz3Logz+Γ3′(1)z−1,z∈C∖(−∞,0] is a Pick function and find its Stieltjes representation.