{"title":"根据具有一定解析函数的凸性的从属条件确定星形和近凸函数的一些性质","authors":"H. Sahin, Ismet Yildiz","doi":"10.37863/umzh.v75i7.7214","DOIUrl":null,"url":null,"abstract":"UDC 517.5\nInvestigation of the theory of complex functions is one of the most fascinating aspects of theory of complex analytic functions of one variable. It has a huge impact on all areas of mathematics. Many mathematical concepts are explained when viewed through the theory of complex functions. Let \n\n f\n \n (\n z\n )\n \n ∈\n A\n ,\n\n \n\n f\n \n (\n z\n )\n \n =\n z\n +\n \n \n ∑\n \n n\n ≥\n 2\n \n ∞\n \n \n \n a\n n\n \n \n z\n n\n \n ,\n\n be an analytic function in the open unit disc normalized by \n\n f\n \n (\n 0\n )\n \n =\n 0\n\n and \n\n f\n '\n \n (\n 0\n )\n \n =\n 1.\n\n For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where \n\n r\n\n is a positive integer of order \n\n \n 2\n \n -\n r\n \n \n\n \n\n \n (\n 0\n <\n \n 2\n \n -\n r\n \n \n ≤\n \n \n \n 1\n 2\n \n \n \n )\n \n .\n\n By using subordination, we propose a criterion for \n\n f\n \n (\n z\n )\n \n ∈\n \n S\n *\n \n \n [\n \n a\n r\n \n ,\n \n b\n r\n \n ]\n \n .\n\n The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study their regional transformations. In addition, the properties of convexity for \n\n f\n \n (\n z\n )\n \n ∈\n A\n\n are examined.","PeriodicalId":163365,"journal":{"name":"Ukrains’kyi Matematychnyi Zhurnal","volume":"64 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function\",\"authors\":\"H. Sahin, Ismet Yildiz\",\"doi\":\"10.37863/umzh.v75i7.7214\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"UDC 517.5\\nInvestigation of the theory of complex functions is one of the most fascinating aspects of theory of complex analytic functions of one variable. It has a huge impact on all areas of mathematics. Many mathematical concepts are explained when viewed through the theory of complex functions. Let \\n\\n f\\n \\n (\\n z\\n )\\n \\n ∈\\n A\\n ,\\n\\n \\n\\n f\\n \\n (\\n z\\n )\\n \\n =\\n z\\n +\\n \\n \\n ∑\\n \\n n\\n ≥\\n 2\\n \\n ∞\\n \\n \\n \\n a\\n n\\n \\n \\n z\\n n\\n \\n ,\\n\\n be an analytic function in the open unit disc normalized by \\n\\n f\\n \\n (\\n 0\\n )\\n \\n =\\n 0\\n\\n and \\n\\n f\\n '\\n \\n (\\n 0\\n )\\n \\n =\\n 1.\\n\\n For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where \\n\\n r\\n\\n is a positive integer of order \\n\\n \\n 2\\n \\n -\\n r\\n \\n \\n\\n \\n\\n \\n (\\n 0\\n <\\n \\n 2\\n \\n -\\n r\\n \\n \\n ≤\\n \\n \\n \\n 1\\n 2\\n \\n \\n \\n )\\n \\n .\\n\\n By using subordination, we propose a criterion for \\n\\n f\\n \\n (\\n z\\n )\\n \\n ∈\\n \\n S\\n *\\n \\n \\n [\\n \\n a\\n r\\n \\n ,\\n \\n b\\n r\\n \\n ]\\n \\n .\\n\\n The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study their regional transformations. In addition, the properties of convexity for \\n\\n f\\n \\n (\\n z\\n )\\n \\n ∈\\n A\\n\\n are examined.\",\"PeriodicalId\":163365,\"journal\":{\"name\":\"Ukrains’kyi Matematychnyi Zhurnal\",\"volume\":\"64 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-07-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Ukrains’kyi Matematychnyi Zhurnal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.37863/umzh.v75i7.7214\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ukrains’kyi Matematychnyi Zhurnal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.37863/umzh.v75i7.7214","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
复变函数理论的研究是单变量复解析函数理论中最吸引人的一个方面。它对数学的各个领域都有巨大的影响。许多数学概念都是通过复函数理论来解释的。设f (z)∈A, f (z) = z +∑n≥2∞A n z n是由f(0) = 0和f '(0) = 1归一化的开单元圆盘上的解析函数。对于接近凸的星形函数，利用从属性质得到了新的不同的条件，其中r是2 - r阶的正整数(0 < 2 - r≤1 2)。利用从属关系，给出了f (z)∈S * [a r, b r]的判据。根据星形函数和近凸函数的从属性质，研究了它们在一定条件下的关系。同时，我们分析了一些解析函数的凸性，并研究了它们的区域变换。此外，还研究了f (z)∈A的凸性性质。
Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function
UDC 517.5
Investigation of the theory of complex functions is one of the most fascinating aspects of theory of complex analytic functions of one variable. It has a huge impact on all areas of mathematics. Many mathematical concepts are explained when viewed through the theory of complex functions. Let
f
(
z
)
∈
A
,
f
(
z
)
=
z
+
∑
n
≥
2
∞
a
n
z
n
,
be an analytic function in the open unit disc normalized by
f
(
0
)
=
0
and
f
'
(
0
)
=
1.
For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where
r
is a positive integer of order
2
-
r
(
0
<
2
-
r
≤
1
2
)
.
By using subordination, we propose a criterion for
f
(
z
)
∈
S
*
[
a
r
,
b
r
]
.
The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study their regional transformations. In addition, the properties of convexity for
f
(
z
)
∈
A
are examined.
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