Arcadii Z. Grinshpan

Completely monotonic functions involving the gamma and q-gamma functions

We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and q-gamma functions.

Inequalities for the gamma and q-gamma functions

We present several sharp inequalities for the classical gamma and q-gamma functions. Some inequalities involve the psi function and its q-analogue. Our results improve, complement, and generalize some known (nonsharp) estimates.

Inequalities generated by the gamma function

We prove sharp inequalities for arbitrary complex vectors and weights generated by the gamma function. Some limiting cases and applications are discussed. They include general combinatorial, exponential, and integral inequalities.

The Bieberbach Conjecture and Milin's Functionals

(i) Bieberbachs conjecture motivated the development of the Loewner parametric method (1923), which became a very useful tool in the theory of univalent functions. (ii In 1971, I. M. Milin constructed a sequence of functionals associated with his exponentiation approach and the Bieberbach conjecture. He conjectured that these functionals were nonpositive, and gave an elementary argument showing that his conjecture implies Bieberbachs. (iii) In 1984, L. de. Branges proved that Milins functionals were nonpositive. He used Loewners method together with results and ideas from several fields of mathematics. This achievement allowed de Branges to confirm the Bieberbach conjecture as a theorem.

The grunsky operator and coefficient differences

We investigate some coefficient properties of univalent functions related to their Grunsky operators and quasiconformal extendibility. We find some bounds of the sharp growth order in n and numerical estimates for coefficient differences of analytic and univalent functions in the unit disk. These results depend on a restriction of the norm of the Grunsky operator.

Estimating the argument of approximate conformal mappings

Let fbe analytic in the unit disk E f(0)=0f(z) ≢ 0 otherwise. We consider the problem of estimating the argument for given bounds for . Such problems arise in measuring the accuracy of approximate conformal mappings of simply connected domains onto the unit disk.

Volterra convolution equations: solution–kernel connection

The authors weighted inequality for two functions is used to estimate a natural connection between solutions and kernels of the first-kind convolution Volterra equations and related convolution integral equations. The solution–kernel estimates and some functionally parameterized solutions lead to various integro-differential, hypergeometric, convolution, exponential, and other inequalities. Also, the obtained estimates allow us to establish certain non-trivial properties of the unknown and complicated solutions. Several examples and applications are discussed.

A family of integral equations with restricted solutions

A family of Volterra integral equations of a special kind is considered. In each equation a given function occurs both in the integrand and in the expression on the right equal to the convolution integral. It is proved that all solutions to the equations of this family satisfy certain sharp mean modulus inequality. This result can be used as a source of various inequalities for special functions. Some examples and applications that are based on the explicit hypergeometric and general series solutions are discussed. They involve the Jacobi and Laguerre polynomials as well as the confluent hypergeometric functions.

Remarks on the Grunsky norm and p th root transformation

Abstract We show that the norm of the Grunsky operator generated by a univalent function does not decrease with a pth root transformation, p⩾2. The result is sharp for each p.

On an identity related to multivalent functions

We prove an algebraic identity by induction. This identity is very important in the coefficient problem for analytic functions that are p-valent in the unit disk.