Uniqueness of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -functions and meromorphic functions under sharing of sets

In [4], the authors proved that the zero set of a uniqueness polynomial, satisfying some additional conditions, becomes a unique range set for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -functions. They also determined the conditions under which a polynomial becomes a strong uniqueness polynomial for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -functions. These results are the improved version of one result of [3]. In this paper we obtain a number of uniqueness theorems for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -functions in the extended Selberg class, which significantly extend the results of [3] and [4] in a new direction and improve them in some cases. From our results we can show some classes of unique range sets for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -functions which cannot be found by the results of [3] and [4].