On a class of functions unifying the classes of Paatero, Robertson and others
We study a class Mkλ(α,β,b,c) of analytic functions which unifies a number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu and others. Thus our class includes convex and starlike functions of order β, spirallike functions of order β and functions for which zf′ is spira...
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| Format: | Article |
| Language: | English |
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Wiley
1988-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171288000304 |
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| author | S. Bhargava S. Nanjunda Rao |
| author_facet | S. Bhargava S. Nanjunda Rao |
| author_sort | S. Bhargava |
| collection | DOAJ |
| description | We study a class Mkλ(α,β,b,c) of analytic functions which unifies a number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu and others. Thus our class includes convex and starlike functions of order β, spirallike functions of order β and functions for which zf′ is spirallike of order β, functions of boundary rotation utmost kπ, α-convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the class Vk of functions of bounded boundary rotation, yield some representation theorems and a variational principle for our class. A consequence of these basic theorems is a theorem for this class Mkλ(α,β,b,c) which unifies some earlier results concerning the radii of convexity of functions in the class Vkλ(β) of Moulis and those concerning the radii of starlikeness of functions in the classes Uk of Pinchuk and U2(β) of Robertson etc. By applying an estimate of Moulis concerning functions in Vkλ(0), we obtain an inequality in the class Mkλ(α,β,b,c) which will contain an estimate for the Schwarzian derivative of functions in the class Vkλ(β) and in particular the estimate of Moulis for the Schwarzian of functions in Vkλ(0). |
| format | Article |
| id | doaj-art-86a5320dda83489fb85b2b14650b0d57 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1988-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-86a5320dda83489fb85b2b14650b0d572025-02-03T00:59:25ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251988-01-0111225125810.1155/S0161171288000304On a class of functions unifying the classes of Paatero, Robertson and othersS. Bhargava0S. Nanjunda Rao1Department of Mathematics, University of Mysore, Manasagangothri, Mysore 570 006, IndiaDepartment of Mathematics, University of Mysore, Manasagangothri, Mysore 570 006, IndiaWe study a class Mkλ(α,β,b,c) of analytic functions which unifies a number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu and others. Thus our class includes convex and starlike functions of order β, spirallike functions of order β and functions for which zf′ is spirallike of order β, functions of boundary rotation utmost kπ, α-convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the class Vk of functions of bounded boundary rotation, yield some representation theorems and a variational principle for our class. A consequence of these basic theorems is a theorem for this class Mkλ(α,β,b,c) which unifies some earlier results concerning the radii of convexity of functions in the class Vkλ(β) of Moulis and those concerning the radii of starlikeness of functions in the classes Uk of Pinchuk and U2(β) of Robertson etc. By applying an estimate of Moulis concerning functions in Vkλ(0), we obtain an inequality in the class Mkλ(α,β,b,c) which will contain an estimate for the Schwarzian derivative of functions in the class Vkλ(β) and in particular the estimate of Moulis for the Schwarzian of functions in Vkλ(0).http://dx.doi.org/10.1155/S0161171288000304convex functionsstarlike functionsspirallike functionsfunctions of bounded boundary rotationα-convex functions. |
| spellingShingle | S. Bhargava S. Nanjunda Rao On a class of functions unifying the classes of Paatero, Robertson and others International Journal of Mathematics and Mathematical Sciences convex functions starlike functions spirallike functions functions of bounded boundary rotation α-convex functions. |
| title | On a class of functions unifying the classes of Paatero, Robertson and others |
| title_full | On a class of functions unifying the classes of Paatero, Robertson and others |
| title_fullStr | On a class of functions unifying the classes of Paatero, Robertson and others |
| title_full_unstemmed | On a class of functions unifying the classes of Paatero, Robertson and others |
| title_short | On a class of functions unifying the classes of Paatero, Robertson and others |
| title_sort | on a class of functions unifying the classes of paatero robertson and others |
| topic | convex functions starlike functions spirallike functions functions of bounded boundary rotation α-convex functions. |
| url | http://dx.doi.org/10.1155/S0161171288000304 |
| work_keys_str_mv | AT sbhargava onaclassoffunctionsunifyingtheclassesofpaaterorobertsonandothers AT snanjundarao onaclassoffunctionsunifyingtheclassesofpaaterorobertsonandothers |