A new connection between geometric function theory and number theory is derived from Ramanujan’s work on modular equations. This connection involves the function recurrent in the theory of plane quasiconformal maps. Ramanujan’s modular identities yield numerous new functional identities for for various primes p.
Some inequalities related to the submultiplicative properties of the distortion function are derived.
For μ ∈ ℂ such that Re μ > 0 let denote the class of all non-vanishing analytic functions f in the unit disk with f(0) = 1 and 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 . In the final section we graphically illustrate the region of variability for several sets of parameters.
The rate of growth of the energy integral of a quasiregular mapping is estimated in terms of a special isoperimetric condition on . The estimate leads to new Phragmén-Lindelöf type theorems.
It is shown that the approximate continuity of the dilatation matrix of a quasiregular mapping f at implies the local injectivity and the asymptotic linearity of f at . Sufficient conditions for to behave asymptotically as are given. Some global injectivity results are derived.
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