We study Volterra–Lévy processeswith kernels regularly varying at infinity. We prove the weak convergence of rescaled processes to Gaussian limits with explicit covariance structures and establish large and moderate deviation principles. The analysis combines regular variation techniques, cumulant methods, and the Gärtner–Ellis theorem, extending classical asymptotic results for Lévy processes to the Volterra framework. The results are obtained for finite-dimensional distributions.
Let S denote the class of univalent functions in the open unit disk 𝔻 = z ∈ ℂ: |z| < 1 of the form f(z)=z+∑_n=2^∞a_nz^n . We establish sharp upper bounds for the second Toeplitz determinant of logarithmic coefficients and the logarithmic coefficients of the inverse functions for the classes of starlike and convex functions. Moreover, we examine the invariance property of our main results.
In the ring of arithmetic functions, several characterizations of multiplicative /completely multiplicative and additive /completely additive functions are established. They are derived by employing concepts related to unitary convolution, including generalized unitary Möbius functions and logarithmic and exponential operators. The distributivity of arithmetic functions through discriminative products is also investigated, leading to unitary analogues of classical results obtained via the Dirichlet convolution. These results provide necessary and/or sufficient conditions for multiplicative/ completely multiplicative, and additive /completely additive behavior, extending earlier studies.
Let ℓ be a nonnegative integer, and let a and b be two relatively prime integers such that 615 ℓ +3 ⩽ a < b. In this note, assuming the generalized Riemann hypothesis, we prove that there exists a prime p ∈ ( ℓ ab, ( ℓ +1)ab − a − b) that has exactly ( ℓ + 1) different expressions of the form p = ax + by, where x and y are nonnegative integers. This result generalizes, in particular, the recent work of Dai, Ding, and Wang, which confirms the 2020 conjecture of Ramírez Alfonsín and Skałba.
We examine the differential subordination related to the geometric and harmonic means. In particular, the discussed differential subordination generalizes the well-known Briot–Bouquet differential subordination. The main results are applicable to construct nontrivial subclasses of the class of standardly normalized holomorphic functions in the unit disk. In addition, a new type of ordinary differential equation is proposed for study.
