Abstract Let $$\mathcal {S}^*(\alpha _1,\alpha _2)$$ S ∗ ( α 1 , α 2 ) , where $$ \alpha _1, \alpha _2 \in (0,1]$$ α 1 , α 2 ∈ ( 0 , 1 ] , represent the class of functions f that are analytic in the open unit disk $$\mathbb {D}$$ D , normalized by $$f(0) = f'(0) - 1=0$$ f ( 0 ) = f ′ ( 0 ) - 1 = 0 , and satisfying the following double-sided inequality: $$\begin{aligned} -\frac{\pi \alpha _1}{2}< \arg \left\{ \frac{zf'(z)}{f(z)}\right\} <\frac{\pi \alpha _2}{2}, \quad (z\in \mathbb {D}). \end{aligned}$$ - π α 1 2 < arg z f ′ ( z ) f ( z ) < π α 2 2 , ( z ∈ D ) . In this manuscript, we estimate the coefficients and logarithmic coefficients associated with functions that belong to the class $$\mathcal {S}^*(\alpha _1,\alpha _2)$$ S ∗ ( α 1 , α 2 ) . As a result, we provide a general bound for the coefficients of a strongly starlike function, which has been an open question until now. Finally, we derive upper and lower bounds for the expression $$\textrm{Re}\{zf'(z)/f(z)\}$$ Re { z f ′ ( z ) / f ( z ) } , where $$f\in \mathcal {S}^*(\alpha _1,\alpha _2)$$ f ∈ S ∗ ( α 1 , α 2 ) .

Abstract Suppose a Lie group G acts on a vertex algebra $$\mathcal {V}$$ V . In this article we construct a vertex algebra $${\tilde{V}}$$ V ~ , which is an extension of $$\mathcal {V}$$ V by a big central vertex subalgebra identified with the algebra of functionals on the space of regular $$\mathfrak {g}$$ g -connections $$(\textrm{d}+A)$$ ( d + A ) . The category of representations of $${\tilde{\mathcal {V}}}$$ V ~ fibres over the set of connections, and the fibres should be viewed as $$(\textrm{d}+A)$$ ( d + A ) -twisted modules of $$\mathcal {V}$$ V , generalizing the familiar notion of g -twisted modules. In fact, another application of our result is that it proposes an explicit definition of $$(\textrm{d}+A)$$ ( d + A ) -twisted modules of $$\mathcal {V}$$ V in terms of a twisted commutator formula, and we feel that this subject should be pursued further. Vertex algebras with big centers appear in practice as critical level or large level limits of vertex algebras. In particular, we have in mind limits of the generalized quantum Langlands kernel, in which case G is the Langland dual and $$\mathcal {V}$$ V is conjecturally the Feigin-Tipunin vertex algebra and the extension $${\tilde{\mathcal {V}}}$$ V ~ is conjecturally related to the Kac-DeConcini-Procesi quantum group with big center. With the current article, we can give a uniform and independent construction of these limits.

Abstract The existence of weighted nontangential limits and growth rates near a polar set E of positive superharmonic functions satisfying a nonlinear inequality $$-\Delta u(x)\le cd(x,E)^{-\beta } u(x)^p$$ - Δ u ( x ) ≤ c d ( x , E ) - β u ( x ) p and having singularities on E are investigated. A main result extends Lions’ result (1980) regarding the asymptotic behavior near an isolated singularity of a positive solution of the Lane–Emden equation to the case of non-isolated singularities, and complements the author and Ono’s result (2014) regarding removable singularities of solutions of semilinear elliptic equations.

Abstract We correct a mistake found in our the article “Cesàro-like operators acting on spaces of analytic functions”, published in Anal. Math. Phys. 12 (2022), no. 2, Paper No. 51, 29 pp.

Abstract In this article we investigate the property of complete monotonicity within a special family $$\mathcal {F}_s$$ F s of functions in s variables involving logarithms. The main result of this work provides a linear isomorphism between $$\mathcal {F}_s$$ F s and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in $$\mathcal {F}_s$$ F s is semi-algebraic. This gives a finite time algorithm to decide whether a function in $$\mathcal {F}_s$$ F s is completely monotone.

In 2008 LEP got an award as the