Complex deformable Rolle, mean value, Flett, and Sahoo-Riedel theorems in the complex plane

In this paper, we proved analogues of Rolle's theorem, the mean value theorem, Flett's theorem, and the Sahoo-Riedel theorem for $\lambda$-deformable analytic functions in the complex plane. The main tool was the identity

$$D_{\lambda }f(z)-\delta f(z)=\lambda f^{\prime }(z),$$

which connects the $\lambda$-complex deformable derivative to the classical complex derivative. A key observation was that the natural vanishing condition in the deformable setting is not $D_{\lambda }f(z)=0$, but $D_{\lambda }f(z)-\delta f(z)=0$. We also characterized constant functions in terms of the complex deformable derivative and proved that the complex deformable Rolle theorem and the complex deformable mean value theorem are equivalent. Unlike Flett's theorem, the Sahoo-Riedel theorem requires no boundary condition on the derivative and thus applies to all $\lambda$-deformable analytic functions. All results reduce to their classical complex counterparts when $\lambda =1$.