Value distribution of meromorphic functions or holomorphic curves and its applications

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(A) For different complex numbers λ_1, … ,λ_<n+1>(n≧2), we investigated the defect relation for the exponential curve fe=[eλ^1^2, …,eλ^<n+1>^2] and we obtained the following theorem. This cannot be obtained from the general theory. Let X be a subset of c^<n+1>-{0} in general position and X^+ be the subset of X the defect of each element of which is positive. Further let d_I is the number of aei in X^+ and D be the convex hull of λ_1, … ,λ_<n+1>.Theorem. If D is n+l-gon, then 0<d_I<1 andΣ__<a∈X^+>∫(a,f_e)≦n+1-Σ^^<n+1>__<I=1>α_I(1-d_I) (α_I>0).(B) Let f be a transcendental holomorphic curve from C into the n-dimensional complex projective space, X be a subset of C^<n+1>-{0} in N-subgeneral position(N>n≧2). Further, we suppose that there exists {a_1, … , a_q} in X satisfying (q≦∞).Σ^^q__<I=1>∫(a_I,f)=2N-n+1.Then, we have the following theorem.Theorem. If N>n=2m (m is a positive integer), then there are [(2N-n+l)/(n+l)]+1 elements in {a_1, … , a_q} satisfying δ(a_j,f)=1.c As applications of value distribution of meromorphic functions, we obtained the followings:(I)We applied the Wiman-Valiron theory to some q-difference equations to obtain the same results on the existence of solutions and the growth of solutions as in the case of differential equations.(ii)We considered meromorphic solutions of the Riccati differential equation with meromorphic coefficients to obtain that the solutions of the equation w' +w^2 +aγ(z)=0 are all one-valued meromorphic in the complex plane. Here, γ(z) is the Weierstrass γ-function with g_3≠0, a=(1-m^2)/4 and m(>2) is integer satisfying m≠6n.