## The set where a transcendental entire function is large and an allied problem |

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### Contents

The set where a transcendental entire function is large | 8 |

The extremal entire functions | 20 |

Appendix to Chapter 3 | 36 |

### Common terms and phrases

ak's angular measure arg(z Baernstein belonging to Qn best possible Calcutta Calcutta University canonical product Caratheodory's theorem Chebychev choose circular symmetrisation compact subsets complement conformal mapping converges uniformly Cornell University define an entire define the desired denote the number desired entire function domain Edrei entire function f(z finite order fixed real number Fuchs and Shea function of finite function of order gn(W harmonic measure Hence holomorphic identity map infinite product inside integer leading coefficient unity Lemma Let f(z lim inf lim sup log+ log+1 f(rei0 Mathematics maximum modulus meromorphic function n1 log number of zeros on|z open arc open set order of g(z Pk(z/Rk polynomial of degree positive sequence Presidency College Proof prove qn(ic qn(w r^oo Riemann mapping theorem Rk,f satisfying sequence of circles sequence of positive Subinoy Chakravarty sufficiently large sufficiently small Sup|pn(f transcendental entire function z/Rk zero order