Area Of The Complement Of The Fast Escaping Sets Of A Family Of Entire Functions And The Meromorphic Solutions Of A Certain Complex Differential Equation

This doctoral thesis consists of four chapters.In the first chapter,we introduce the research background and the main results.Our results are mainly divided into two aspects.In the first section,we study the area of the complement of the fast escaping sets of a family of entire functions.Let f be an entire function with the form f?z?= P?e^z?/e^z,where P is a polynomial with deg?P?? 2 and P?0?? 0.We prove that the Lesbegue area of the complement of the fast escaping set?hence the Fatou set?of f in a horizontal strip of width 2? is finite.We give also a specific formula of the upper bound of Area?S?A?f?c?in terms of the coefficients of the polynomial P,where S is any horizontal strip of width 2? and A?f?c is the complement of the fast escaping set of f.In particular,the corresponding result can be applied to the sine family? sin?z + ??,where ??0and ??C.Finally,our method can be adopted also to the type of entire functions with the form g?z?=P?w?/wm o exp?z?,completely similarly,where m>1 is a positive integer,P is a polynomial with degree deg?P?>m+1 and P?0?? 0.In the second section,we consider the cubic Briot-Bouquet differential equation with the form a₁ f'³ + a₂f'²f+a₃f'f² + a₄f³ + a₅f'² + a₆f'f + a₇f₂+a₈f'+a₉f+a₁0=0,where a_j?C?j = 1,...,10?are constants,and we give the explicit forms of its all possible meromorphic solution.In the second chapter,we mainly introduce the basic knowledge of the dynamics of transcendental entire functions,and give some results of the value distribution theory and basic properties of Fatou sets and Julia sets of transcendental entire functions.We also introduce several special transcendental entire functions,such as the entire functions of bounded singular type and finite singular type.Finally,we introduce some results about the escaping sets and the fast escaping sets.In the third chapter,we give the proof of the results about the area of the com-plement of the fast escaping sets of a family of entire functions.In the fourth chapter,we give the proof of the theorems about the meromorphic solutions of cubic Briot-Bouquet differential equation.