On Two Trigonometric Inequalities of Carslaw and Gasper

Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-...
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Autor*in:	Alzer, Horst [verfasserIn] Kwong, Man Kam

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	Trigonometric polynomials inequalities

Anmerkung:	© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Übergeordnetes Werk:	Enthalten in: Results in mathematics - Berlin : Springer, 1978, 77(2022), 3 vom: 05. Mai
Übergeordnetes Werk:	volume:77 ; year:2022 ; number:3 ; day:05 ; month:05

Links:	Volltext

DOI / URN:	10.1007/s00025-022-01660-1

Katalog-ID:	SPR046908501

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520			\|a Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible.
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912			\|a GBV_ILN_60
912			\|a GBV_ILN_62
912			\|a GBV_ILN_63
912			\|a GBV_ILN_65
912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2056
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
912			\|a GBV_ILN_2143
912			\|a GBV_ILN_2144
912			\|a GBV_ILN_2147
912			\|a GBV_ILN_2148
912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
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912			\|a GBV_ILN_4037
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912			\|a GBV_ILN_4125
912			\|a GBV_ILN_4126
912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
912			\|a GBV_ILN_4305
912			\|a GBV_ILN_4306
912			\|a GBV_ILN_4307
912			\|a GBV_ILN_4313
912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
912			\|a GBV_ILN_4324
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912			\|a GBV_ILN_4326
912			\|a GBV_ILN_4328
912			\|a GBV_ILN_4333
912			\|a GBV_ILN_4334
912			\|a GBV_ILN_4335
912			\|a GBV_ILN_4336
912			\|a GBV_ILN_4338
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allfields	10.1007/s00025-022-01660-1 doi (DE-627)SPR046908501 (SPR)s00025-022-01660-1-e DE-627 ger DE-627 rakwb eng Alzer, Horst verfasserin (orcid)0000-0001-6970-061X aut On Two Trigonometric Inequalities of Carslaw and Gasper 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible. Trigonometric polynomials (dpeaa)DE-He213 inequalities (dpeaa)DE-He213 Kwong, Man Kam aut Enthalten in Results in mathematics Berlin : Springer, 1978 77(2022), 3 vom: 05. Mai (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:77 year:2022 number:3 day:05 month:05 https://dx.doi.org/10.1007/s00025-022-01660-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2022 3 05 05
spelling	10.1007/s00025-022-01660-1 doi (DE-627)SPR046908501 (SPR)s00025-022-01660-1-e DE-627 ger DE-627 rakwb eng Alzer, Horst verfasserin (orcid)0000-0001-6970-061X aut On Two Trigonometric Inequalities of Carslaw and Gasper 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible. Trigonometric polynomials (dpeaa)DE-He213 inequalities (dpeaa)DE-He213 Kwong, Man Kam aut Enthalten in Results in mathematics Berlin : Springer, 1978 77(2022), 3 vom: 05. Mai (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:77 year:2022 number:3 day:05 month:05 https://dx.doi.org/10.1007/s00025-022-01660-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2022 3 05 05
allfields_unstemmed	10.1007/s00025-022-01660-1 doi (DE-627)SPR046908501 (SPR)s00025-022-01660-1-e DE-627 ger DE-627 rakwb eng Alzer, Horst verfasserin (orcid)0000-0001-6970-061X aut On Two Trigonometric Inequalities of Carslaw and Gasper 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible. Trigonometric polynomials (dpeaa)DE-He213 inequalities (dpeaa)DE-He213 Kwong, Man Kam aut Enthalten in Results in mathematics Berlin : Springer, 1978 77(2022), 3 vom: 05. Mai (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:77 year:2022 number:3 day:05 month:05 https://dx.doi.org/10.1007/s00025-022-01660-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2022 3 05 05
allfieldsGer	10.1007/s00025-022-01660-1 doi (DE-627)SPR046908501 (SPR)s00025-022-01660-1-e DE-627 ger DE-627 rakwb eng Alzer, Horst verfasserin (orcid)0000-0001-6970-061X aut On Two Trigonometric Inequalities of Carslaw and Gasper 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible. Trigonometric polynomials (dpeaa)DE-He213 inequalities (dpeaa)DE-He213 Kwong, Man Kam aut Enthalten in Results in mathematics Berlin : Springer, 1978 77(2022), 3 vom: 05. Mai (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:77 year:2022 number:3 day:05 month:05 https://dx.doi.org/10.1007/s00025-022-01660-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2022 3 05 05
allfieldsSound	10.1007/s00025-022-01660-1 doi (DE-627)SPR046908501 (SPR)s00025-022-01660-1-e DE-627 ger DE-627 rakwb eng Alzer, Horst verfasserin (orcid)0000-0001-6970-061X aut On Two Trigonometric Inequalities of Carslaw and Gasper 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible. Trigonometric polynomials (dpeaa)DE-He213 inequalities (dpeaa)DE-He213 Kwong, Man Kam aut Enthalten in Results in mathematics Berlin : Springer, 1978 77(2022), 3 vom: 05. Mai (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:77 year:2022 number:3 day:05 month:05 https://dx.doi.org/10.1007/s00025-022-01660-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 77 2022 3 05 05
language	English
source	Enthalten in Results in mathematics 77(2022), 3 vom: 05. Mai volume:77 year:2022 number:3 day:05 month:05
sourceStr	Enthalten in Results in mathematics 77(2022), 3 vom: 05. Mai volume:77 year:2022 number:3 day:05 month:05
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Trigonometric polynomials inequalities
isfreeaccess_bool	false
container_title	Results in mathematics
authorswithroles_txt_mv	Alzer, Horst @@aut@@ Kwong, Man Kam @@aut@@
publishDateDaySort_date	2022-05-05T00:00:00Z
hierarchy_top_id	327046066
id	SPR046908501
language_de	englisch
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We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Trigonometric polynomials</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">inequalities</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Kwong, Man Kam</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Results in mathematics</subfield><subfield code="d">Berlin : Springer, 1978</subfield><subfield code="g">77(2022), 3 vom: 05. 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author	Alzer, Horst
spellingShingle	Alzer, Horst misc Trigonometric polynomials misc inequalities On Two Trigonometric Inequalities of Carslaw and Gasper
authorStr	Alzer, Horst
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topic_title	On Two Trigonometric Inequalities of Carslaw and Gasper Trigonometric polynomials (dpeaa)DE-He213 inequalities (dpeaa)DE-He213
topic	misc Trigonometric polynomials misc inequalities
topic_unstemmed	misc Trigonometric polynomials misc inequalities
topic_browse	misc Trigonometric polynomials misc inequalities
format_facet	Elektronische Aufsätze Aufsätze Elektronische Ressource
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title	On Two Trigonometric Inequalities of Carslaw and Gasper
ctrlnum	(DE-627)SPR046908501 (SPR)s00025-022-01660-1-e
title_full	On Two Trigonometric Inequalities of Carslaw and Gasper
author_sort	Alzer, Horst
journal	Results in mathematics
journalStr	Results in mathematics
lang_code	eng
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publishDateSort	2022
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author_browse	Alzer, Horst Kwong, Man Kam
container_volume	77
format_se	Elektronische Aufsätze
author-letter	Alzer, Horst
doi_str_mv	10.1007/s00025-022-01660-1
normlink	(ORCID)0000-0001-6970-061X
normlink_prefix_str_mv	(orcid)0000-0001-6970-061X
title_sort	on two trigonometric inequalities of carslaw and gasper
title_auth	On Two Trigonometric Inequalities of Carslaw and Gasper
abstract	Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
abstractGer	Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
abstract_unstemmed	Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
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title_short	On Two Trigonometric Inequalities of Carslaw and Gasper
url	https://dx.doi.org/10.1007/s00025-022-01660-1
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doi_str	10.1007/s00025-022-01660-1
up_date	2024-08-28T05:54:57.959Z
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fullrecord_marcxml	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR046908501</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20240827161849.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">220506s2022 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00025-022-01660-1</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR046908501</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s00025-022-01660-1-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Alzer, Horst</subfield><subfield code="e">verfasserin</subfield><subfield code="0">(orcid)0000-0001-6970-061X</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">On Two Trigonometric Inequalities of Carslaw and Gasper</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2022</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)%$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}%$and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)%$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}%$are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)%$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}%$and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).%$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}%$The constants %$3\pi /4%$ and 64/75 are best possible.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Trigonometric polynomials</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">inequalities</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Kwong, Man Kam</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Results in mathematics</subfield><subfield code="d">Berlin : Springer, 1978</subfield><subfield code="g">77(2022), 3 vom: 05. 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On Two Trigonometric Inequalities of Carslaw and Gasper

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