Wyniki wyszukiwania

1

Singular values, Ramanujan modular equations, and Landen transformations

100%

Vuorinen M.

Studia Mathematica

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1996

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tom 121

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nr 3

221-230

EN

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.

2

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

63%

-L. Qiu S., Vuorinen M.

Studia Mathematica

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1995-1996

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tom 117

|

nr 3

225-242

EN

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

3

On local injectivity and asymptotic linearity of quasiregular mappings

51%

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

Studia Mathematica

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1998

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tom 128

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nr 3

243-271

EN

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.

4

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

51%

Ponnusamy S., Vasudevarao A., Vuorinen M.

Colloquium Mathematicum

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2009

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tom 116

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nr 1

31-46

EN

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.

Ograniczanie wyników

3 Studia Mathematica

1 Colloquium Mathematicum

4 Vuorinen M.

1 -L. Qiu S.

1 Martio O.

1 Ponnusamy S.

1 Ryazanov V. I.

1 Vasudevarao A.

1 Ya. Gutlyanskiĭ V.

1 2009

1 1998

2 1996

Singular values, Ramanujan modular equations, and Landen transformations

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

On local injectivity and asymptotic linearity of quasiregular mappings

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