Summary: For a positive integer , applying the Schwarz lemma for analytic functions in the open unit disk , an assertion on a lemma well-known as Jack’s lemma, proved by S. S. Miller and P. T. Mocanu [J. Math. Anal. Appl. 65, 289–305 (1978; Zbl 0367.34005)], is given. Further, using a method from the proof of a subordination relation discussed by T. J. Suffridge [Duke Math. J. 37, 775–777 (1970; Zbl 0206.36202)] and T. H. MacGregor [J. Lond. Math. Soc., II. Ser. 9, 530–536 (1975; Zbl 0331.30011)], some differential subordination property concerning the subordination
for functions and which are analytic in , is deduced, and an extension of some subordination relation is given.