A Half-Discrete Mulholland-Type Inequality Involving One Multiple Upper Limit Function

In this paper, by means of weight coefficient methods and the Euler–Maclaurin summation formula, a half-discrete Mulholland-type inequality involving one multiple upper limit function is established, with the kernel (x +nβ)−λ (λ > 0 , β ∈(0,1]). In comparison with the existing result with the kernel (x +nβ)−λ, the homogeneous-type kernel is replaced by a logarithmic (Mulholland-type) one, the admissible parameter ranges are enlarged from λ ∈ (0,5 − m], m ∈ {0,1,2,3,4}, to λ > 0 with arbitrary m ∈ ℕ 0, and the partial sum is removed. The best constant factor of the obtained inequality is \(\frac{\text{$\Gamma$}(\lambda + m)}{\beta^{1/p}\text{$\Gamma$}(\lambda)}B(\lambda_{1} + m,\lambda_{2}) \). Several equivalent conditions for the best possible constant are further provided, together with some particular cases and extended forms; in particular, for m = 0 with specific parameters, the classical Mulholland inequality is recovered.