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The paper studies logarithmic convexity and concavity of the generalizedhypergeometric function with respect to simultaneous shift of severalparameters. We use integral representations and properties of Meijer's $G$function to prove log-convexity. When all parameters are shifted we use seriesmanipulations to examine the power series coefficients of the generalizedTur\'{a}nian formed by the generalized hypergeometric function. In cases whenall zeros of the generalized hypergeometric function are real, we furtherexplore the consequences of the extended Laguerre inequalities and formulate aconjecture about reality of zeros.

Log-concavity and Tur\'{a}n-type inequalities for the generalized hypergeometric function