# On the Zeros of the Differential Polynomial \(\varphi(z)f^{2}(z)f^{\prime}(z)^{2}-1\)
## 1 Introduction and Results  
In this paper, we assumed that the reader is familiar with the notations of Nevanlinna theory (see, e.g., [1, 2]).Let \(f(z)\) and \(\alpha(z)\) be two meromorphic functions in the complex plane. If \(T(r,\alpha)=S(r,f)\), then \(\alpha(z)\) is called a small function of \(f(z)\). **Definition 1**. : _Reference [2] Let \(k\) be a positive integer. For any constant \(a\) in the complex plane we denote by \(N_{k}(r,1/(f-a))\) the counting function of those \(a\)-points of \(f\) whose multiplicities are not great than \(k\), by \(N_{k}(r,1/(f-a))\) the counting function of those \(a\)-points of \(f\) whose multiplicities are not less than \(k\), by \(N_{k}(r,1/(f-a))\) the counting function of those \(a\)-points of \(f\) with multiplicity \(k\), and denote the reduced counting function by \(\overline{N}_{k})(r,1/(f-a))\), \(\overline{N}_{(k}(r,1/(f-a))\) and \(\overline{N}_{k}(r,1/(f-a))\), respectively._  
**Definition 2**. : _If \(z_{0}\) is a pole of \(f(z)\) with multiplicity \(l\), then we say \(\omega(f,z_{0})=l\), \(\overline{\omega}(f,z_{0})=1\). Otherwise, \(\omega(f,z_{0})=\overline{\omega}(f,z_{0})=0\)._  
Clearly, for \(p\) meromorphic functions, we have  
\[\omega(\Pi_{j=1}^{p}f_{j},z_{0})\leq\sum_{j=1}^{p}\omega(f_{j},z_{0}), \tag{1}\]  
and when \(f_{j}\
eq 0(\forall j=1,2,...,p)\), we have  
\[\omega(\Pi_{j=1}^{p}f_{j},z_{0})=\sum_{j=1}^{p}\omega(f_{j},z_{0}). \tag{2}\]  
**Definition 3**. : _Reference [2] Let \(f(z)\) be a transcendental meromorphic function. The deficiency of a complex number \(a\) with respect to \(f(z)\) is defined by_  
\[\delta(a,f)=\liminf_{r\to\infty}\frac{m(r,\frac{1}{f-1})}{T(r,f)}=1-\limsup_{ r\to\infty}\frac{N(r,\frac{1}{f-a})}{T(r,f)}.\]  
It is easy to see \(0\leq\delta(a,f)\leq 1\). Definition 4: Reference [2] If the coefficients of differential polynomials \(M[f]\) are \(a_{j}\), \(j=0,1,...,n\), which satisfy \(m(r,a_{j})=S(r,f)\), then \(M[f]\) is called a quasi-differential polynomials in \(f\). In 1959, [PERSON] proved the following theorem. Theorem 1: (see [3]) Let \(f\) be a transcendental meromorphic function, \(n(\geq 3)\) be an integer. Then \(\phi=f^{n}f^{\prime}\) has infinitely many zeros for finite non-zero complex value \(a\). Moreover, [PERSON] [4] conjectured that Theorem 1 remains valid for the cases \(n=1,2\). In 1979, [PERSON] [5] confirmed the case \(n=2\) and the conjecture was proved by [PERSON] [6] in 1995 and independently by [PERSON] and [PERSON] [7]. Naturally, we will ask that if the constant \(a\) is replaced by a small function of \(\alpha(z)\), what is the distributions of zeros of \(ff^{\prime}-\alpha\)? Many scholars have studied this problem. In 1994, [PERSON] proved the following two results:  
Theorem 2: (see [8]) Let \(f\) be a transcendental meromorphic function, \(\alpha(\
ot\equiv 0,\infty)\) is a small function and \(\delta(\infty;f)>\frac{7}{9}\), then \(ff^{\prime}-\alpha\) has infinitely many zeros. Theorem 3: (see [8]) Let \(f\) be a transcendental meromorphic function, \(a(\
ot\equiv 0,\infty)\) is a small function and \(2\delta(0;f)+\delta(\infty;f)>1\), then \(ff^{\prime}-\alpha\) has infinitely many zeros. In 1997, [PERSON] proved the following special case when \(f\) is of finite order and \(\alpha\) is a polynomial:  
Theorem 4: (see [9]) If \(f\) is a transcendental meromorphic function of finite order and \(\alpha\) is a non-vanishing polynomial, then \(ff^{\prime}-\alpha\) has infinitely many zeros. In order to achieve the desired result, there are some conditions for the zeros or poles of \(f\) in Theorem 2 and Theorem 3. Except for the order of \(f\), there is no other conditional constraint in Theorem 4, but the result is only valid for the polynomial. [PERSON] deals with the general situation of the small functions and proved the following result:  
Theorem 5: (see [10]) Let \(f\) be a transcendental meromorphic function and \(\alpha(\
ot\equiv 0,\infty)\) be a small function. Then \(ff^{\prime}-\alpha\) and \(ff^{\prime}+\alpha\) at least one has infinitely many zeros.