## On meromorphic solutions of certain nonlinear differential equations.(English)Zbl 1047.34101

The authors deal with differential equations of the form $L(f)+ p(z, f)= h(z),\tag{1}$ where $$L(f)$$ is a linear differential polynomial in $$f$$ with meromorphic coefficients, $$p(z, f)$$ is a polynomial in $$f$$ with meromorphic coefficients, and $$h(z)$$ is meromorphic. Define $$L_f:= \{h$$ meromorphic: $$T(r,h)= S(r,h)\}$$ and denote by $$F$$ the family of meromorphic solutions to (1) such that, whenever $$f\in F$$, all coefficients in (1) are in $$L_f$$, and $$N(r\cdot f)= S(r\cdot f)$$. It follows that, if $$f,g\in F$$, then $T(r\cdot g)= O(T(r\cdot f))+ S(r\cdot f).$ Moreover, if $$\alpha> 1$$, then, for some $$r_\alpha> 0$$, $T(r\cdot g)= O(T(\alpha r,f))$ for all $$r\geq r_\alpha$$. The authors show that, if $$f$$ is a meromorphic solution to (1) such that all coefficients in (1) are in $$L_f$$, then $$\rho(f)\geq \rho(h)$$. If $$n=: \deg_f p(z, f)\geq k+2$$ and $$N(r\cdot f)= S(r\cdot f)$$ then $$\rho(f)= \rho(f)$$ and $$\mu(f)= \mu(f)$$.
For the equation $L(f)- p(z) f^n= h(z),$ $$h(z)$$ be a meromorphic function, the authors show that the method used by Yang can be modified to obtain similar uniqueness results for meromorphic solutions to this generalized equation, when $$n\geq 4$$.

### MSC:

 34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
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### References:

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