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Singular values, Ramanujan modular equations, and Landen transformations

100%

Vuorinen M.

Studia Mathematica

1996

tom 121

nr 3

221-230

A new connection between geometric function theory and number theory is derived from Ramanujan's work on modular equations. This connection involves the function $φ_K(r)$ recurrent in the theory of plane quasiconformal maps. Ramanujan's modular identities yield numerous new functional identities for $φ_{1/p}(r)$ for various primes p.

Submultiplicative properties of the $φ_{K}$-distortion function

63%

-L. Qiu S. , Vuorinen M.

Studia Mathematica

1995-1996

tom 117

nr 3

225-242

Some inequalities related to the submultiplicative properties of the distortion function $φ_K(r)$ are derived.

Region of variability for spiral-like functions with respect to a boundary point

51%

Ponnusamy S. , Vasudevarao A. , Vuorinen M.

Colloquium Mathematicum

2009

tom 116

nr 1

31-46

For μ ∈ ℂ such that Re μ > 0 let $ℱ_{μ}$ denote the class of all non-vanishing analytic functions f in the unit disk 𝔻 with f(0) = 1 and $Re(2π/μ zf'(z)/f(z) + (1+z)/(1-z)) > 0$ in 𝔻. For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ 𝔻̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class $ℱ_{μ}(λ) = {f ∈ ℱ_{μ}: f'(0) = (μ/π)(λ - 1) and f''(0) = (μ/π)(a(1-|λ|²) + (μ/π)(λ-1)² - (1-λ²))}$. In the final section we graphically illustrate the region of variability for several sets of parameters.

On local injectivity and asymptotic linearity of quasiregular mappings

51%

Ya. Gutlyanskiĭ V. , Martio O. , Ryazanov V. I. , Vuorinen M.

Studia Mathematica

1998

tom 128

nr 3

243-271

It is shown that the approximate continuity of the dilatation matrix of a quasiregular mapping f at $x_0$ implies the local injectivity and the asymptotic linearity of f at $x_0$. Sufficient conditions for $log|f(x) - f(x_0)|$ to behave asymptotically as $log|x - x_0|$ are given. Some global injectivity results are derived.