New extensions for a theorem by Mocanu

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For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=f"(0)=0, P. T. Mocanu (Mathematica (Cluj), 42(2000)) has considered some sufficient arguments of f'(z)+zf"(z) for |(zf'(z)/f(z))|<πμ/2. The object of the present paper is to discuss those probrems for f(z) with f"(0)=f"'(0)=...=f⁽ⁿ⁾(0)=0 and f⁽ⁿ⁺¹⁾(0) ≠ 0.