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Singular values, Ramanujan modular equations, and Landen transformations

M. Vuorinen — 1996

Studia Mathematica

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.

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

S. PonnusamyA. VasudevaraoM. Vuorinen — 2009

Colloquium Mathematicae

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

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