Denoting by \(x_{n,k}(\alpha,\beta)\) and \(x_{n,k}(\lambda)= x_{n,k} (\lambda-1/2, \lambda-1/2)\) the zeros, in decreasing order, of the Jacobi polynomial \(P_n^{(\alpha,\beta)} (x)\) and of the ultraspherical (or Gegenbauer) polynomial \(C_n^\lambda(x)\), respectively, the authors investigate the monotonicity of \(x_{n,k}(\alpha,\beta)\) as functions of \(\alpha\) and \(\beta\) \((\alpha,\beta >-1)\). Necessary conditions such that the zeros of \(P_n^{(a,b)} (x)\) are smaller (greater) than the zeros of \(P_n^{(\alpha, \beta)}(x)\) are provided. \textit{A. Markoff} [Math. Ann. 27, 177-182 (1886; JFM 18.0069.02)] proved that \(x_{n,k}(a,b) x_{n,k} (\alpha, \beta))\) for every \(n\in\mathbb{N}\) and each \(k(1\leq k\leq n)\) if \(a>\alpha\) and \(b\beta)\). The present authors prove the converse statement of Markov's theorem. The question of how large the function \(f_n( \lambda)\) could be such that the products \(f_n(\lambda)x_{n,k} (\lambda)\) \((k=1, \dots, [n/2])\) are increasing functions of \(\lambda\), for \(\lambda>-1/2\), is also discussed. \textit{Á. Elbert} and \textit{P. D. Siafarikas} [J. Approximation Theory 97, No. 1, 31-39 (1999; Zbl 0923.33004)] proved that \(f_n(\lambda)= \{\lambda+ (2n^2+1)/(4n+2)\}^{1/2}\) obeys this property. The paper under review establishes the sharpness of their result.