For any real polynomial $p(x)$ of even degree k, Shapiro [‘Problems around polynomials: the good, the bad and the ugly $\ldots $ ’, Arnold Math. J.1(1) (2015), 91–99] proposed the conjecture that the sum of the number of real zeros of the two polynomials $(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)$ and $p(x)$ is larger than 0. We prove that the conjecture is true except in one case: when the polynomial $p(x)$ has no real zeros, the derivative polynomial $p{'}(x)$ has one real simple zero, that is, $p{'}(x)=C(x)(x-w)$ , where $C(x)$ is a polynomial with $C(w)\ne 0$ , and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has no real zeros.

For any real polynomial p(x) of even degree k. B. Shapiro propose the 12th conjecture saying that the sum of the number of real zeros of two polynomials ((k - 1)(p'(x))2 - kp(x)p''(x) and p(x)) is larger than 0. We comprehensively prove the original B.Shapiro's 12th conjecture and B.Shapiro's 12th conjecture of entire functions by showing all cases that the conjecture are true, and the case that it is not true. Mathematics Subject Classification: Primary 30C15 Secondary 30D20, 26C10

The main result of this paper shows a totally new necessary and sufficient condition to determine both real and complex zeros of derivative of all entire and meromorphic functions of one complex variable in the extended complex plane. By using the theorem, we reprove some results about zeros of derivative of Xi function, Gamma function and digamma function in a new way.