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Abstract:

Let Υ denote the class of functions χ(ξ) of the form χ(ξ)=ξ+∑_(n=2)^∞▒a_n ξ^n which are analytic in the open unit disc Δ=\{ ξ∈C: |ξ|<1 }. In recent times investigating the properties of several existing and new subclasses of quasi-convex functions have gained importance and attracted researchers working in the theory of univalent functions. Using the Carlson-Shaffer operator, we introduce new subclasses of quasi-convex functions. The coefficient bounds for functions belonging to the defined function classes are our main results. Further, we establish various well-known results as corollaries to our main results.

Keywords:

Analytic function,quasi-convex,close to convex,close to star,Janowski function,coefficient estimates,Carlson-Shaffer operator,

Refference:

I. Altıntas, O., and Kılıc, Ӧ.Ӧ. (2018). Coefficient estimates for a class containing quasi-convex functions, Turkish Journal of Mathematics, 42(5), 2819-2825. 10.3906/mat-1805-90.
II. Carlson, B. C., and Shaffer, D. B. (1984). Starlike and prestarlike hypergeometric functions, SIAM Journal on Mathematical Analysis, 15(4), 737-745. 10.1137/0515057.
III. Friedland, S. and Schiffer, M. (1976). Global results in control theory with applications to univalent functions, Bulletin of the American Mathematical Society, 82(6), 913-915. 10.1090/S0002-9904-1976-14211-5.
IV. Goodman, A. W. (1972). On close-to-convex functions of higher order, Annales Universitatis Scientiarum Budapestinensis de Rolando Eőtvős Nominatae. Sectio Mathematica, 15, 17-30.
V. Janowski, W. (1973). Some extremal problems for certain families of analytic functions, Annales Polonici Mathematici, 28, 297-326. 10.4064/ap-28-3-297-326.
VI. Kaplan, W. (1952). Close-to-convex schlicht functions, Michigan Mathematical Journal, 1, 169-185.
VII. Karthikeyan, K.R., Amarender Reddy, K., and Murugusundaramoorthy, G. (2022). On classes of Janowski functions associated with a conic domain, Italian Journal of Pure and Applied Mathematics, Vol.-47, 684-698.
VIII. Karthikeyan, K. R., Varadharajan, S., and Lakshmi, S. (2024). Some properties of close-to-convex functions associated with a strip domain, Sahand Communications in Mathematical Analysis, 21(2), 105-124. 10.22130/scma.2023.1987052.1233.
IX. Mehrok, B. S., and Singh, G. (2010). A subclass of close-to-convex functions, International Journal of Mathematical Analysis (Ruse), 4, 1319-1327.
X. Mehrok, B. S., Singh, G., and Gupta, D. (2010). A subclass of analytic functions, Global Journal of Mathematical Sciences: Theory and Practical, 2(1), 91-97.
XI. Mehrok, B. S., and Singh, G. (2012). A subclass of close-to-star functions, International Journal Modern Mathematical Sciences, 4(3), 139-145.
XII Mukhamadullina, G. I., Kornev, K. G., and Alimov, M. M. (1999). Hysteretic effects in the problems of artificial freezing, SIAM Journal on Applied Mathematics, 59(2), 387- 410. 10.1137/S0036139996313782.
XIII. Noor, K. I., and Thomas, D. K. (1980). Quasiconvex univalent functions, International Journal of Mathematics and Mathematical Sciences, 3(2), 255-266. 10.1155/S016117128000018X.
XIV. Noor, K. I. (1987). On quasiconvex functions and related topics, International Journal of Mathematics and Mathematical Sciences, 10(2), 241-258. 10.1155/S0161171287000310.
XV. Prokhorov, D. V. (2001). Coefficients of holomorphic functions, Journal of Mathematical Sciences (New York), 106 (6), 3518-3544. 10.1023/A:1011975914158.
XVI. Reade, M. O. (1955). On close-to-convex univalent functions, Michigan Mathematical Journal, 3, 59-62.
XVII. Rensaa, R. J. (2003). Univalent functions and frequency analysis, Rocky Mountain Journal of Mathematics, 33(2), 743-758. 10.1216/rmjm/1181069976.
XVIII. Singh, G. (2013). Certain subclasses of α-close-to-star functions, International Journal Modern Mathematical Sciences, 7(1), 80-89. 10.1007/s10013-013-0032-4.
XIX. Srivastava, H. M., Altıntas, O. and Serenbay, S. K. (2011). Coefficient bounds for certain subclasses of starlike functions of complex order, Applied Mathematics Letters, 24(8), 1359-1363. 10.1016/j.aml.2011.03.010.
XX. Vasil’ev, A. (2001). Univalent functions in the dynamics of viscous flows, Computational Methods and Function Theory, 1(2), 311-337. 10.1007/BF03320993.
XXI Vasil’ev, A. and Markina, I. (2003). On the geometry of Hele-Shaw flows with small surface tension. Interfaces Free Bound. 5 (2), 183-192. 10.4171/IFB/77.
XXII. Xiong, L., and Liu, X. (2012). A general subclass of close-to-convex functions Kragujevac Journal of Mathematics, 36(2), 251-260.

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Abstract:

Within this paper, we create the ideas of slightly Σα-shelter, shortly sΣα - shelter, and slightly Σα-compact, shortly Σα-compact fuzzy sets. In addition, we have established certain theorems concerning our vision in fuzzy topological spaces, fuzzy subspaces, fuzzy continuity mapping, fuzzy open mapping, fuzzy T₂-spaces (Hausdorff spaces), and their different characterizations. Finally, we have discussed our vision about the “good extension” characteristics

Keywords:

fuzzy sets,fuzzy topological spaces,sΣα-shelter,sΣα-compact,

Refference:

I. Abbas, S. E. “On intuitionistic fuzzy compactness.” Information Sciences 173.1-3 (2005): 75-91. https://doi.org/10.1016/j.ins.2004.07.004
II. Aygün, Halis. “α-compactness in L-fuzzy topological spaces.” Fuzzy Sets and Systems 116.3 (2000): 317-324. 10.1016/S0165-0114(98)00345-5
III. Benchalli, S. S., and G. P. Siddapur. “On the level spaces of fuzzy topological spaces.” Bulletin of Mathematical Analysis and Applications 1.2 (2009): 57-65. https://eudml.org/doc/229234
IV. Chadwick, J. J. “A generalised form of compactness in fuzzy topological spaces.” Journal of Mathematical Analysis and Applications 162.1 (1991): 92-110. 10.1016/0022-247X(91)90180-8
V. Chang, Chin-Liang. “Fuzzy topological spaces.” Journal of mathematical Analysis and Applications 24.1 (1968): 182-190. 10.1016/0022-247X(68)90057-7
VI. Chen, Yixiang. “On compactness of induced I (L)-fuzzy topological spaces.” Fuzzy Sets and Systems 88.3 (1997): 373-378. 10.1016/S0165-0114(96)00074-7
VII. Dongsheng, Zhao. “The N-compactness in L-fuzzy topological spaces.” Journal of Mathematical Analysis and Applications 128.1 (1987): 64-79. 10.1016/0022-247x(87)90214-9
VIII. EL-Shafei, M. E., I. M. Hanafy, and A. I. Aggour. “RS-Compactness in L-fuzzy topological spaces.” Kyungpook Mathematical Journal 42.2 (2002): 417-428.
IX. Gantner, T. E., R. C. Steinlage, and R. H. Warren. “Compactness in fuzzy topological spaces.” Journal of Mathematical Analysis and Applications 62.3 (1978): 547-562. 10.1016/0022-247X(78)90148-8
X. Georgiou, D. N., and B. K. Papadopoulos. “On fuzzy compactness.” Journal of Mathematical Analysis and Applications 233.1 (1999): 86-101. 10.1006/jmaa.1999.6268
XI. Guojun, Wang. “A new fuzzy compactness defined by fuzzy nets.” Journal of Mathematical Analysis and Applications 94.1 (1983): 1-23. 10.1016/0022-247X(83)90002-1
XII. Hong, Woo Chorl. “RS-compact spaces.” Journal of the Korean Mathematical Society 17.1 (1980): 39-43.
XIII. Katsaras, A. K. “Ordered fuzzy topological spaces.” Journal of Mathematical Analysis and Applications 84.1 (1981): 44-58. 10.1016/0022-247X(81)90150-5
XIV. Kim, J. K. “α-compact fuzzy topological spaces.” Korean Journal of Computational and Applied Mathematics 1 (1994): 79-84. https://doi.org/10.1007/BF02943052
XV. Kudri, S. R. T. “Compactness in L-fuzzy topological spaces.” Fuzzy Sets and Systems 67.3 (1994): 329-336. 10.1016/0165-0114(94)90260-7
XVI. Kudri, S. R. T., and M. W. Warner. “RS-compactness in L-fuzzy topological spaces.” Fuzzy Sets and Systems 86.3 (1997): 369-376. 10.1016/S0165-0114(95)00405-X
XVII. Li, Hong-Yan, and Fu-Gui Shi. “Degrees of fuzzy compactness in L-fuzzy topological spaces.” Fuzzy sets and systems 161.7 (2010): 988-1001. 10.1016/j.fss.2009.10.012
XVIII. Lipschutz, Seymour. “General Topology, Schaum’s Outline Series.” (1965).
XIX. Lowen, R. “Fuzzy topological spaces and fuzzy compactness.” Journal of Mathematical analysis and applications 56.3 (1976): 621-633. 10.1016/0022-247X(76)90029-9
XX. Lowen, Robert. “Initial and final fuzzy topologies and the fuzzy Tychonoff theorem.” Journal of Mathematical Analysis and Applications 58.1 (1977): 11-21. 10.1016/0022-247X(77)90223-2
XXI. Lowen, Robert. “A comparison of different compactness notions in fuzzy topological spaces.” Journal of Mathematical Analysis and Applications 64.2 (1978): 446-454. 10.1016/0022-247X(78)90052-5
XXII. Lowen, R. “Compactness properties in the fuzzy real line.” Fuzzy Sets and Systems 13.2 (1984): 193-200. 10.1016/0165-0114(84)90019-8
XXIII. Mahbub, Aman, Sahadat Hossain, and M. Altab Hossain. “Some properties of compactness in intuitionistic fuzzy topological spaces.” International Journal of Fuzzy Mathematical Archive 16.2 (2018): 39-48. DOI: 10.22457/ijfma.v16n1a6
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XXV. Meenakshi, Arunachalam R., and Muniasamy Kaliraja. “g-inverses of interval valued fuzzy matrices.” Journal of Mathematical & Fundamental Sciences 45.1 (2013): 83-92. 10.5614/j.math.fund.sci.2013.45.1.7
XXVI. Murdeshwar, Mangesh G. “General topology, Wiley” (1983).
XXVII. Noiri, Takashi. “On RS-compact spaces.” Journal of the Korean Mathematical Society 22.1 (1985): 19-34.
XXVIII. Pao-Ming, Pu, and Liu Ying-Ming. “Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence.” Journal of Mathematical Analysis and Applications 76.2 (1980): 571-599. 10.1016/0022-247X(80)90048-7
XXIX. Saha, Apu Kumar, and Debasish Bhattacharya. “Countable fuzzy topological space and countable fuzzy topological vector space.” Journal of Mathematical & Fundamental Sciences 47.2 (2015): 154-166. 10.5614/j.math.fund.sci.2015.47.2.4
XXX. Shi, Fu-Gui. “A new form of fuzzy -compactness.” Mathematica Bohemica 131.1 (2006): 15-28.
XXXI. Shi, Fu-Gui. “A new definition of fuzzy compactness.” Fuzzy sets and systems 158.13 (2007): 1486-1495. 10.1016/j.fss.2007.02.006
XXXII. Talukder, M. A. M., and D. M. Ali. “Certain features of partially -compact fuzzy sets.” Journal of Mechanics of Continua and Mathematical Sciences 9.1 (2014):1322-1340. 10.26782/jmcms.2014.07.00005
XXXIII. Warner, M. W., and R. G. McLean. “On compact Hausdofff L-fuzzy spaces.” Fuzzy Sets and Systems 56.1 (1993): 103-110. 10.1016/0165-0114(93)90190-S
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XXXVII. Ying, Mingsheng. “Compactness in fuzzifying topology.” Fuzzy Sets and Systems 55.1 (1993): 79-92. 10.1016/0165-0114(93)90303-Y.
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Abstract:

In this paper, the issues of developing a calculation method for the formation of cracks normal to the longitudinal axis of bent reinforced concrete structures of circular cross-section without prestressing the reinforcement based on a nonlinear deformation model using a bilinear diagram of the state of concrete are considered. The prerequisites based on which theoretical dependences are constructed to determine the complex internal forces of a round normal section before the formation of cracks is presented. Based on stereometry, dependencies are presented to determine the forces in the concrete and reinforcement of the compressed zone and the forces in the stretched zone, respectively, for concrete and reinforcement. Since it is difficult to analytically solve a system of equations with negative and positive exponents, it is recommended to carry out numerical iterative processes to determine the desired unknowns: heights of the compressed zone, maximum stresses in concrete and reinforcement of the compressed zone. Numerical studies have made it possible to determine the value of the elastic-plastic moment of the annular section and to identify its dependence on the strength of concrete.

Keywords:

Circular cross-section,moment of crack formation,neutral axis,arc length,sector area,elastic-plastic moment of resistance,

Refference:

I. Bronshteyn Ilya N., and Semendyayev Konstantin A. Spravochnik po matematike. Dlya inzhenerov i uchashchikhsya VTUZov. – Moskva, «Nauka».1986. S.544. https://www.geokniga.org/books/10582
II. Golyshev Aleksander B., and Bachinskiy Vladimir Y., and Polishchuk Vitaliy P., and Kharchenko Aleksander V. Proyektirovaniye zhelezobetonnykh konstruktsiy. Spravochnoye posobiye. – Kiyev, Budivel’nyk, 1990. https://dwg.ru/lib/1280
III. Karpenko Sergei N. Obshchiy metod rascheta zhelezobetonnykh elementov kol’tsevogo secheniya. // Trudy 1-y Vserossiyskoy konferentsii «Beton na rubezhe tret’yego tysyacheletiya», 2001. https://www.elibrary.ru/item.asp?ysclid=m7vkidp113126215346&edn=slhznp)
IV. Krylov Yuriy K., and Polyakov Igor D. Proletnyye stroyeniya iz tsentrofugirovannogo zhelezobetona. Transportnoye stroitel’stvo. – M., №6, 1970.
V. Kudzis Aleksey P. Zhelezobetonnyye konstruktsii kol’tsevogo secheniya. – Vil’nyus, Mintis, 1975.
VI. Lastochkin Vasiliy G., and Petsol’d Timofei M., and Tarasov Vladimir V. Opyt proizvodstva i primeneniya tsentrofugirovannykh konstruktsiy v promyshlennom stroitel’stve. Obzornaya informatsiya. – Minsk, BelNIINTI, 1985.
VII. Morshteyn, Oleg B. Izgotovleniye tsentrofugirovannykh kolonn s konsolyami dlya promzdaniy. Transportnoye stroitel’stvo. – №12, 1978, https://ais.by/article/opyt-primeneniya-centrifugirovannyh-lineynyh-elementov-s-poperechnymi-secheniyami
VIII. Mukhamediyev Tahir A. Uchet neuprugikh svoystv betona pri raschete zhelezobetonnykh konstruktsiy po obrazovaniyu treshchin. – Stroitel’naya mekhanika i raschet sooruzheniy, № 5, 2018. https://www.elibrary.ru/item.asp?id=35606774&ysclid=m7vlj8gx5n354667764)
IX. Parfenov Sergei G., and Okusok Sergei A. Primeneniye deformatsionnykh modeley dlya rascheta treshchinostoykosti zhelezobetonnykh elementov. V sb. «Fundamental’nyye poiskovyye i prikladnyye issledovaniya RAASN po nauchnomu obespecheniyu razvitiya arkhitektury, gradostroitel’stva i stroitel’noy otrasli Rossiyskoy Federatsii». Tom 2. – Moskva, PAO «T8 Izdatel’skiye Tekhnologii, 2017.
X. Pastushkov Gennadiy P. Unifitsirovannyye karkasnyye zdaniya s tsentrofugirovannymi kolonnami. Obzornaya informatsiya. – Minsk, BelNIINTI, 1988, 52. http://unicat.nlb.by/opac/pls/pages.view_doc?off=0&siz=20&qid=75987&format=full&nn=11)
XI. Petsol’d Timofei M., and Pastushkov Gennadiy P.,. Karkasy proizvodstvennykh zdaniy iz unifitsirovannykh prednapryazhennykh tsentrofugirovannykh elementov. Beton i zhelezobeton. – M., №4, 1990. https://www.booksite.ru/beton/1990/1990_4.pdf
XII. Trekin Nikolay N., and Kodysh Emil N., and Mamin Aleksander N., and Trekin Dmitriy N.,. Improving methods of evaluating the crack resistance of concrete structures. Proceedings 2nd International Workshop «Durability and Sustainability of Concrete Structures (DSCS-2018)», Moscow, Russia. – American Concrete Institute, 2018.
XIII. Trekin Nikolay N., and Kodysh Emil N., and Trekin Dmitriy N. Raschet po obrazovaniyu normal’nykh treshchin na osnove deformatsionnoy modeli. – Promyshlennoye i grazhdanskoye stroitel’stvo, №7, 2016. https://www.elibrary.ru/item.asp?edn=whkjxn&ysclid=m7vlv124h1711223019
XIV. Trekin Nikolay N., and Kodysh Emil N., and Trekin Dmitriy N. Sovershenstvovaniye metoda otsenki treshchinostoykosti izgibayemykh zhelezobetonnykh elementov. – Beton i zhelezobeton, №1, 2020. https://www.elibrary.ru/item.asp?id=44294660&ysclid=m7vlta0tvl531195817
XV. Trekin Nikolay N., and Kodysh Emil N., and Andryan Konstantin R. Treshchinostoykost’ zhelezobetonnykh konstruktsiy kruglykh secheniy. – Academia. Arkhitektura i stroitel’stvo (3). 10.22337/2077-9038-2022-3-104-109
XVI. Zalesov Aleksandr S., and Mukhamediyev Tahir A., and Chistyakov Eugeniy A.. Raschet treshchinostoykosti zhelezobetonnykh konstruktsiy po novym normativnym dokumentam. – Beton i zhelezobeton, №5, 2002. https://www.elibrary.ru/item.asp?id=30748751&ysclid=m7vkmw3xp7950354133
XVII. Zalesov Aleksandr S., and Chistyakov Eugeniy A.. Deformatsionnaya raschetnaya model’ zhelezobetonnykh elementov pri deystvii izgibayushchikh momentov i prodol’nykh sil – Beton i zhelezobeton, №5, 1996. https://www.elibrary.ru/item.asp?edn=xmvqdb&ysclid=m7vkwkelug868968971

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