Abstract
For a positive integer $n$, applying Schwarz's lemma related to analytic functions $w(z)=c_n z^n+\cdots$ in the open unit disk $\mathbb{U}$, some assertion in a certain lemma which is well-known as Jack's lemma proven by Miller and Mocanu [J. Math. Anal. Appl. 65 (1978), 289-305] is given. Further, by using a certain method of the proof of subordination relation which was discussed by Suffridge [Duke Math. J. 37 (1970), 775--777] and MacGregor [J. London Math. Soc. (2) 9 (1975), 530-536], some differential subordination property concerning with the subordination $$ p(z)\prec q(z^n)\qquad(z\in\mathbb{U}) $$ for functions $p(z)=a+a_n z^n+\cdots$ and $q(z)=a+b_1 z+\cdots$ which are analytic in $\mathbb{U}$ is deduced, and an extension of some subordination relation is given.
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