If M 2m is a closed Kähler spin manifold of positive scalar curvature R, then each eigenvalue λ of type r ( r ∈ {1, …, [( m + 1)/2]}) of the Dirac operator D satisfies the inequality λ 2 ≥ rR 0/4 r − 2, where R 0 is the minimum of R on M 2 m . Hence, if the complex dimension m is odd (even) we have the estimation λ≥ (m+1) R 0 4m (λ≥) mR 0 4(m−1) for the first eigenvalue of D. In the paper is also considered the limiting case of the given inequalities. In the limiting case with m = 2 r − 1 the manifold M 2 m must be Einstein. The manifolds S 2, S 2 × S 2, S 2 × T 2, P 3( C ), F( C ), P 3( C ) × T 2 and F( C 3) × T 2, where F( C 3) denotes the flag manifold and T 2 the 2-dimensional flat torus, are examples for which the first eigenvalue of the Dirac operator realizes the limiting case of the corresponding inequality. In general, if M 2 m is an example of odd complex dimension m, then M 2 m × T 2 is an example of even complex dimension m + 1. The limiting case is characterized by the fact that here appear eigenspinors of D 2 which are Kählerian twistor-spinors.