Strict Monotonicity of the First Positive Steklov Eigenvalue on Eccentric Annuli

We prove strict monotonicity of the first positive Steklov eigenvalue on planar eccentric annuli with fixed inner and outer radii. Starting from a double-center holomorphic representation of harmonic functions, we use Taylor and Laurent coefficient transformations to reduce the Steklov problem to two fixed-point equations on sequence spaces. We show that the associated operators are positive and Hilbert–Schmidt, and that the even operator is a positive rank-one perturbation of the odd operator. Using the spectral theory of irreducible compact positive operators, we identify the first positive Steklov eigenvalue through a spectral-radius equation. Monotonicity of the operator entries then yields strict decrease of this eigenvalue with increasing eccentricity, resolving Conjecture 6.1 of Nigam.