Does this complex integral transform have a name?

Question

We are all familiar with the Stieltjes transform of a real function of bounded variation $F$, which is defined as follows (up to some sign convention that varies across authors): $$\forall z\in\mathbb{C}^+\qquad s_F(z)=\int_{-\infty}^{+\infty}\frac{1}{x-z}dF(x)~.$$ My question is whether there is a name for the closely related transformation: $$\forall z\in\mathbb{C}^+\qquad m_F(z)=\int_{-\infty}^{+\infty}\frac{x}{x-z}dF(x)~.$$ Technically, we could call it "the Stieltjes transform of the incomplete moment function of order one", but that's a bit of a mouthful. The two transforms are related by the equation: $$\forall z\in\mathbb{C}^+\qquad m_F(z)=1+z\,s_F(z)~.$$ I tried to look up the Generalized Stieltjes transform, but it came out as something completely different, namely: $$\forall z\in\mathbb{C}^+\qquad s_F^\rho(z)=\int_{-\infty}^{+\infty}\frac{1}{(x-z)^\rho}dF(x)\quad\text{for some index }\rho>0~.$$