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The investigation of logarithmic coefficients in the theory of univalent functions began with Milin, who demonstrated their importance for understanding geometric features of these mappings through their connection with the Taylor coefficients h m . If denotes the family of univalent functions on the unit disk with the expansion the logarithmic coefficients γ m ( h ) are given by Bazilevi (1965) later underlined the role of these coefficients in several coefficient‐related problems for univalent functions. In the present paper, we introduce a novel technique for obtaining bounds on | γ m ( h )| when h belongs to the Bazilevi class of type ( α , β ) and order γ . The proposed approach refines earlier estimates and extends known results in this area.

A number of important observables exhibit logarithms in their perturbative description that are induced by emissions at widely separated rapidities. These include transverse-momentum (qT) logarithms, logarithms involving heavy-quark or electroweak gauge boson masses, and small-x logarithms. In this paper, we initiate the study of rapidity logarithms, and the associated rapidity divergences, at subleading order in the power expansion. This is accomplished using the soft collinear effective theory (SCET). We discuss the structure of subleading-power rapidity divergences and how to consistently regulate them. We introduce a new pure rapidity regulator and a corresponding MS¯\\[ \\overline{\\mathrm{MS}} \\]-like scheme, which handles rapidity divergences while maintaining the homogeneity of the power expansion. We find that power-law rapidity divergences appear at subleading power, which give rise to derivatives of parton distribution functions. As a concrete example, we consider the qT spectrum for color-singlet production, for which we compute the complete qT2/Q2 suppressed power corrections at Oαs\\[ \\mathcal{O}\\left({\\alpha}_s\\right) \\], including both logarithmic and nonlogarithmic terms. Our results also represent an important first step towards carrying out a resummation of subleading-power rapidity logarithms.

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

Darmon, Lauder, and Rotger conjectured that the relative tangent space of an eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of$p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.

Let \\(\\{B_H(t);t\\ge 0\\}\\) be a fractional Brownian motion of order \\(H\\in (0,1)\\), and \\(J_{m,\\alpha}(B_H)\\) be the \\(m\\)-fold weighted integrals of \\(B_H\\) defined as $$ J_{m,\\bm\\alpha}(B_H)(t) =\\int_0^ts_m^{-\\alpha_m}\\int_0^{s_m}\\cdots s_2^{-\\alpha_2}\\int_0^{s_2}s_1^{-\\alpha_1}B_H(s_1)d s_1\\; ds_2\\cdots d s_m, $$ where \\(\\alpha_1+\\cdots+\\alpha_i

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