GROWTH OF A CLASS OF COMPOSITE ENTIRE FUNCTIONS

Jianwu Sun

Abstract

           In this paper, we obtain the following results: Let f₁, f₂ and g₁, g₂ be four transcendental entire functions with T(r, f₁ ) = O^*((log r)^ν e^{(log r)α}) and T(r, g₁ ) = O^*((log r)^β) (i.e., there exist four positive constants K₁, K₂, K₃ and K₄ such that

K1  ≤

T(r, f1  )   ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ (log r)ν e(log r)α

≤  K2 and K3  ≤

T(r, g1  )   ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ (log r)β

≤  K4  ).

If T(r, f₁ ) ∼ T(r, f₂ ), T(r, g₁ ) ∼ T(r, g₂ )    (r → ∞), then T(r, f₁(g₁)) ∼ T(r, f₂(g₂ ))   (r → ∞ , r E) where ν > 0, 0 < α < 1, β > 1 and αβ < 1 and E is a set of finite logarithmic measure. We solved a problem due to C. C. Yang concerning the characteristic functions of the composite functions.