Let f k ( z ) = z + ∑ n = 2 k a n z n {f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( z ) / f ′ ( z ) } {\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\} , where f ( z ) f\left(z) belongs to the subclass J p , q m ( μ , α , β ) {{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean ( p , q ) \left(p,q) -differential operator. In addition, the inclusion relations involving N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.

中文翻译：

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（p，q）-微分算子的解析函数的部分和和包含关系

令fk（z）= z + ∑ n = 2 kanzn {f} _ {k} \ left（z）= z + {\ sum} _ {n = 2} ^ {k} {a} _ {n} {z } ^ {n}是解析函数f（z）= z + ∑ n = 2∞anznf \ left（z）= z + {\ sum} _ {n = 2} ^ {\ infty}的部分和的序列。 {a} _ {n} {z} ^ {n}。在本文中，我们确定Re {f（z）/ fk（z）} {\ rm {Re}} \ {f \ left（z）\ hspace {-0.08em} \ text {/} \ hspace {-0.08em} {f} _ {k} \ left（z）\}，Re {fk（z）/ f（z）} {\ rm {Re}} \ {{f} _ {k} \ left（z）\ hspace {-0.08em} \ text {/} \ hspace {-0.08em} f \ left（z）\}，Re {f'（z）/ fk'（z）} {\ rm { Re}} \ {{f} ^ {^ {\ prime}} \ left（z）\ hspace {-0.08em} \ text {/} \ hspace {-0.08em} {f} _ {k} ^ {^ {\ prime}} \ left（z）\}和Re {fk'（z）/ f'（z）} {\ rm {Re}} \ {{f} _ {k} ^ {^ {\ prime} } \ left（z）\ hspace {-0.08em} \ text {/} \ hspace {-0.08em} {f} ^ {^ {\ prime}} \ left（z）\}，其中f（z）f \ left（z）属于子类J p，qm（μ，α，β）{{\ mathcal {J}}} _ {p，q} ^ {m} \ left（\ mu，\ alpha，\ beta）由Sălăgean（p，q）\ left（p， q）-微分运算符。另外，考虑了涉及该广义函数类别的Nδ（e）{N} _ {\ delta} \ left（e）的包含关系。