Coefficient Estimates for Dirichlet Series

Exact Estimates for Integrals Involving Dirichlet Series with Nonnegative Coefficients

We consider the Dirichlet series ∞ ∑ k=2 akk −1−x =: f(x), x > 0, with coefficients ak ≥ 0 for all k. Among others, we prove exact estimates of certain weighted Lp-norms of f on the unit interval (0, 1) for any 0 < p <∞, in terms of the coefficients ak . Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for int...

Coefficient Estimates for Univalent Functions

Coefficient estimates for Ruscheweyh derivatives

are classes of starlike and strongly starlike functions of order β (0 < β ≤ 1), respectively. Note that S∗(β)⊂ S∗ for 0< β< 1 and S∗(1)= S∗ [5]. Kanas [2] introduced the subclass R̄δ(β) of function f ∈ S as the following. Definition 1.1. For δ ≥ 0, β ∈ (0,1], a function f normalized by (1.1) belongs to R̄δ(β) if, for z ∈D−{0} and Dδf(z)≠ 0, the following holds: ∣∣∣arg z ( Dδf(z) )′ Dδf(z) ∣∣∣≤ βπ...

Dirichlet Series

This definition could have been given to an 18th or early 19th century mathematical audience, but it would not have been very popular: probably they would not have been comfortable with the Humpty Dumpty-esque redefinition of multiplication. Mathematics at that time did have commutative rings: rings of numbers, of matrices, of functions, but not rings with a “funny” multiplication operation def...

Dirichlet Series

where the an are complex numbers and s is a complex variable. Such functions are called Dirichlet series. We call a1 the constant term. A Dirichlet series will often be written as ∑ ann −s, with the index of summation understood to start at n = 1. Similarly, ∑ app −s runs over the primes, and ∑ apkp −ks runs over the prime powers excluding 1. (Not counting 1 as a prime power in that notation is...