Bulletin of the Malaysian Mathematical Sciences Society

We show that all non-trivial solutions of complex differential equation $$f''+ A(z)f'+B(z)f = 0$$ are of infinite order if coefficients A(z) and B(z) are of special type and establish a relation between the hyper-order of these solutions and the orders of coefficients A(z) and B(z). We have also extended these results to higher order complex differential equations.

In this paper, we considered Ricci semi-symmetric real hypersurface in complex two-plane Grassmannians. Then we prove the non-existence of Ricci semi-symmetric Hopf hypersurfaces in complex two-plane Grassmannians by using the method of simultaneous diagonalization for pairwise commutative matrices.

Take three integers $$m\ge 0,\,k\ge 1$$ , and $$n\ge 2$$ . Let $$a\ (\not \equiv 0)$$ be a holomorphic function in a domain $$D$$ of $$\mathbb {C}$$ such that multiplicities of zeros of $$a$$ are at most $$m$$ and divisible by $$n+1$$ . In this paper, we mainly obtain the following normality criterion: Let $${{{\fancyscript{F}}}}$$ be the family of meromorphic functions on $$D$$ such that multiplicities of zeros of each $$f\in {{\fancyscript{F}}}$$ are at least $$k+m$$ and such that multiplicities of poles of $$f$$ are at least $$m+1$$ . If each pair $$(f,g)$$ of $${{\fancyscript{F}}}$$ satisfies that $$f^{n}f^{(k)}$$ and $$g^{n}g^{(k)}$$ share $$a$$ (ignoring multiplicity), then $${{\fancyscript{F}}}$$ is normal.