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Landau–Kolmogorov constants on the real line $C(n,k)$
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Numbers
$n$
$k$ 
$C(n,k)$
2
1:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
comment: $C(2,1)=\sqrt{2}$, the sharp real-line Landau inequality.
3
1:
1.040041911525952057265028412178942693168902670186631054848795540100053155698634386803028318395378743
comment: $C(3,1)=(9/8)^{1/3}$. Its decimal expansion is [5].
3
2:
1.442249570307408382321638310780109588391869253499350577546416194541687596829997339854755479705645257
comment: $C(3,2)=3^{1/3}$.
4
1:
1.080960123845627515188080150636545649237577074725523438013566442592759909790668572506848181127070762
comment: $C(4,1)=(512/375)^{1/4}$.
4
2:
1.095445115010332226913939565601604267905489389995966508453788899464986554245445467601716872327741252
comment: $C(4,2)=\sqrt{6/5}$.
4
3:
1.480165608984570501133579932327673638598123582612376236649724811831493373159923052408883903803767973
comment: $C(4,3)=(24/5)^{1/4}$.
5
1:
1.044257909309795143445369617155702583080420804202537207757613415800232588800623578874460201119220279
5
2:
1.116645971103809882645715451073153178966512006697404016456342160608179528648522296846460026224549912
5
3:
1.119423731735107611629711082081261041249985567058607086520982799131542292296904515252628659613085229
5
4:
1.496277869738844738508102139329782553317006247093254103087568639503680097204500433745703581090839640
6
1:
1.042975934759424987713224444268385291304875959181407035092022454591988122030909376633409060903927837
6
2:
1.080008574702556575951875606723242328822660546749517776644475201185118153624145709698460644913078823
6
3:
1.145196668627736435824330729897501950999839619652348296246234492448250416266178617464720053404835190
6
4:
1.138424476916702582518046240281571132609680214074048728525461304483546940186525453927370175862741075
6
5:
1.508923110643284290550191719873148070862171236105013262152794453130792712479846172257486753868372306
7
1:
1.034512828815729701488329249333596692339745139620939357327899319707667109217458659054233262216955630
7
2:
1.072887507137353595664805891659506220261056541890119140402223915392865083480166754494549789200996332
7
3:
1.104717220735669847567501570575264429123821397214659019581914233758717862584819660097008792558232064
7
4:
1.164791026816302640806601702340784360501913638838375790210818471245444410622867585316580994445572368
7
5:
1.151373382045461313729995172688584990819043920245630811954513161399997960148532873865777865901193501
7
6:
1.517480399903380177908938710297761225260489095996023173203726178328784341069828165974189669725671221
8
1:
1.030861390135361864713241595789456336150068272719826098976711304608500287657563040926484566056341169
8
2:
1.061818777131340074554097594216182248300417086024015758715112684784644812409370144896532584952020040
8
3:
1.096435163853644296935707526508561729005029176289346772667865758667344663448350944273912822631000847
8
4:
1.124072019995686982549146229016352423936859908661053999286395147669285642271791985728333633915503411
8
5:
1.180063225386654524797609769379581604737408941600863302668143359520660449038866649694173619051653140
8
6:
1.161415698186839723180694597715599467694651083536340474328354203342898769937414536573233391978834300
8
7:
1.524083789157826928154696046104474164151266440517726395555233759565802994126093049452905039459320554
9
1:
1.027136704315476751288263664021578129619462523016672317196161769780189843478753036788165325971562831
9
2:
1.055297699992385318592943631482533351608057700792916001110192519298736834504424634032238900420966495
9
3:
1.083356984817429222456651469909822103570802553512392923613927080956459888866104612023167163696363755
9
4:
1.114937726077049190281981382806549693927917088142719069969841988981800836166779587497429248349137475
9
5:
1.139221734644939741643708141803187895104608840810154213574105133060654319995908161600086054506638092
9
6:
1.191971495322854714682295210660451865805612719000950140308978848343090127556760176921894763868451580
9
7:
1.169216011511101266774914727568207634183999267696250267220636826140382324538378137214936648671495837
9
8:
1.529193258885940333110250732384838057607630613025221844752161494579733240976244654370927567020399216
10
1:
1.024473222569173006145971959938149134620849338908748242046267544658732125253923579273323564696288285
10
2:
1.049450903156580733182982259412936221043790495943643302151045383801560083384106706907783634342779406
10
3:
1.075330920354618923761760030374381846287391343330272375197684143067647017636022833264096934429716787
10
4:
1.100961153025894606448419387971566752986760967737052458355388364022906474296137173778289634063700042
10
5:
1.130015196417341302708450173033051177635098830923872630285831591550601002555790041806149187008868025
10
6:
1.151529850576285670462082565651938184079048782482727108247749162859332679445833012607544068986314583
10
7:
1.201617026917570952362868401856371417899395622187271997744899384183923908534704592730260339455345538
10
8:
1.175515129928808191753955053399008223585665079835787020035819404602929061781395080414745326191018457
10
9:
1.533306968567486940157787334710777999821521365577062824793461496458513477045388453661467418210656430
11
1:
1.022195809032082553749189347140145118793945551790701254569140190335233891855048373075998402156963956
11
2:
1.044915804106514750250359784531248143209465329740075067531639931974120641203442077602873783036408668
11
3:
1.068044634133650886027039903126858378158329569532686024086923580260728758772271472895082758316921081
11
4:
1.091983311037279285912710143198889238485304849340154050652180388990475516227455225119917808404573606
11
5:
1.115558765317197341801041434333495132801709962444438526898771749668185099323417499680913342068424939
11
6:
1.142487170598878546117908091951932339562050506037088609984987803483949254614761938499245285237630733
11
7:
1.161686222985487644945001846218866808812440394581146593771173292778959007525862698776300806339103157
11
8:
1.209556897769008085851328159014789489232492294788462178089072752375964967511040580088926568889684178
11
9:
1.180687709592415963137228500649543940001075006809337374314216420036546854084013647330060241785285411
11
10:
1.536676745182548770607292865504011335961969018485658519141393517385978327992402106789821412240616694
12
1:
1.020336865433369235345233606264748610031573642261519293718669729813478933776742920154750134273401116
12
2:
1.041076884038213439887646817160877952597909268680752786995671640647596659077838707994990154040794262
12
3:
1.062270533315680308198127193576883901162096190549111089267726674615653909670295889724286512125953095
12
4:
1.083798057983048287341224723894390910529411273453994265036068160306398742732519099462118829829032295
12
5:
1.106063590182887064307641755282363048279402496195149600408931459987613489955462697630833884720458962
12
6:
1.127876836340324520092384647157044692078784851013458946396701093595203368759323312015336026575449303
12
7:
1.152990385091103839085048694217796971245473472651800325148752082046308977658407510825860116570109101
12
8:
1.170222170726331476845509711431192346913858264939140786447688267168036807286476834628440882919747014
12
9:
1.216216566317048716058061605521104610289101347340918117093748206328230550166066203872622914460672028
12
10:
1.185017554668114514600783141266948163301962656968801459200484335408445707525778691294383521492659689
12
11:
1.539491834774133847273880073416107918058760846462278683002926444554994537122137373091824530131420824
13
1:
1.018754765294425698891203580981044283664832187228957484929252552873230905325904998497041267867655669
13
2:
1.037864746492582647718943216626630179262099495158916944725685013821417960967195507562407334226133631
13
3:
1.057322598323337313047855163449966560261688055092360665255539671902785558754451931715703702868430613
13
4:
1.077177750939741274034738616292425031418923711096700765583451224899509069779136776963202417277128026
13
5:
1.097306968655015975506492254691412826547998935923079055101426373758864425261172552120984780689133014
13
6:
1.118117369176418031399945358219123901121032324318019793229897928615861570970725379068530676979114698
13
7:
1.138404238561620268791905231845593879720755136414536418255218035290051294566203268550486034335122678
13
8:
1.161951607696377742246978866568729256078734123154168728064037051442069255527253588713767929392698074
13
9:
1.177492653719801178463310738003801386057091585362832695010591259479079294542453435980348275774883958
13
10:
1.221879357926434203092136533095689809822296119211024057206276362548642040586851675662254132719360835
13
11:
1.188693057296020979059333592776808507018239049795955905301852097662372670681646515277867229093207228
13
12:
1.541877464194882756935050416108491443967890954180741721265041529163694342777044607463876960297886461
14
1:
1.017404604654913479811143636317562606257246784912708626629231161903367440556633879133596756950918692
14
2:
1.035110975882629910163042763089843033335418665523495854071477976547675885299988843441323384656079374
14
3:
1.053129025209203979743384411650818752546110057879821355548832209527597906299124954341844030015687006
14
4:
1.071449973142441257262355284630581600113354278123678546723280311797082036712318067533460553256561672
14
5:
1.090122541025292378870324221512858592428880205078531717647641686292226018646772329580768978379928453
14
6:
1.109020679291188140495806233700307485085393104184138446664446136294439632245472639538396911622749529
14
7:
1.128554307181898632450898841503567669992682429714915817792268187568663629352871395663608546141982641
14
8:
1.147506447295654212094649669127243561086780287763908336351672315650234547308479445143213038435466213
14
9:
1.169688534921654404160628161753803178986667985837381584217932345917680848776181158328870157391662140
14
10:
1.183760814934514314564302923927480005896864867940112348211030964370939568080002193849401747641231899
14
11:
1.226754453122790049076718442722306658372436084694493411878324851714615136875908003698075738899702968
14
12:
1.191852750100203098691144039280720229406792928564037110724785428785127115745999161454445368457305229
14
13:
1.543925354478125732351204028036534617565976027842613735111189598368188551522208196593798219668840583
15
1:
1.016234578649588495901066722281798081979924142001947005373284821064089466509457466255904376452597058
15
2:
1.032733102809045163465409627598335895939537877770865660777513245169858372645660969289394010483151936
15
3:
1.049498310064343182838252052211141955750626683056487582118417539719517780927505468618166925386691569
15
4:
1.066539251422985972890103063578038286361840746422841395185116930662752422286757720594091593665702411
15
5:
1.083846026791583976281164800925512296535545354126416617905381100536170996109573444775802076843438828
15
6:
1.101466878528998950672735525660123613553574765656038680449425189427316421300503093697449159124497189
15
7:
1.119273438290439711164473131042689153151970450475408178941944205543631392736145097704295720560567413
15
8:
1.137678227349060142368527925801168532324964842309140818838661921196602538704955828763514315985484132
15
9:
1.155453704392426113933241275356050967801351375650889309116768041488851901311625852927423888877722500
15
10:
1.176435385120443318115254414851667570039944143674569895904476140414820515983448243691835286710916974
15
11:
1.189220087541607644334823073097343629316140825224680786343873375385053123273454362659336828543164075
15
12:
1.230995174854335973973153126259421932745653730046317046075398560834390270877299035591279623769259748
15
13:
1.194597884248054899928257555133252891794246199408425191876908219765671397266107874026915934934346038
15
14:
1.545702354431832325355323859798145685379807213990448764507587050293549889683909451216106220734668464
16
1:
1.015212357883350109048606514299988808326378774376654383545392388829814656112945318144774213296244789
16
2:
1.030656003925837790052119747509714211630109776073975223180761932160585866962285015497027592670874665
16
3:
1.046334971319810221082659143388488156265963296897748764028673071252802515638367410126807273143556580
16
4:
1.062251272783483915894154089338294361458032627456652551745996817251335946402719873363013533070299983
16
5:
1.078413295153288119368120201015831685251969884002084397328715567365590400795250687512433967882205359
16
6:
1.094810247199623500861480190716557382227682012521978863181148618120872031151656950448783465168809516
16
7:
1.111490051098685892018635020262841072160026161783276247769983688232711772952215805420687382018128309
16
8:
1.128322396148455177352259288670943312262913294514587802837244921718378589267672099347480074270548427
16
9:
1.145722208229978049933102331694508830595692054296977199125475156748788439335663855298750892489126393
16
10:
1.162452745448753605558608973261519931526810546204557595979413627303379187374763773837546174708535061
16
11:
1.182370840709110052913699149224128870093378969028297818829487108528082991511090442535754643921489945
16
12:
1.194017637171708376883042586419095342650820009520806099085903221856582009951117843011156512427350110
16
13:
1.234717858764691126873676631716026962720567290198988372282560341613467143237612981993670395178802974
16
14:
1.197005081278291839256697673076142420777380724190160326796867717731282624417257529550047499122223563
16
15:
1.547258919042505852089901121032218231490517627469755027789303784419700558939439524569042795239164446
17
1:
1.014311102024903606668475662208370786463833050024702157751928499245647121773885676567776258522649332
17
2:
1.028827054184648748369567372648136121436055200549114376187883765670149574743469729710241056496636635
17
3:
1.043550616954310949239683170720025588811796287974855011088826581080828118623170017999546332421547150
17
4:
1.058485282440270652807904792543330293714097709692933280906950675334650281006708817984408038203846240
17
5:
1.073632487204159248363076832376625074661834007600489467400494934169429354679507792212788685129042963
17
6:
1.089000098359689703633456314083075368093634406471274910276639245205916367492797763409941684240227857
17
7:
1.104576604892590288245451477445081709559568059308253463967480465909544490402236534891504944031700567
17
8:
1.120409720156026699239847763699422536542553742213845776175945086875361616433774587876607357935324414
17
9:
1.136367483347260424787492185361836311828855063910149927911065559584170078192073140318405657148091358
17
10:
1.152867037706335857386458218268752704597441237875729461895046577169837638409025038403086020673974052
17
11:
1.168663552247125569549004962121594819120104016481126337334804701529739386662173508763078146997009507
17
12:
1.187632856784627478993059670897160459439759081115266145370413324908944985690199795181841658642810996
17
13:
1.198266827760402819073896671979382153560803763428032748843064855124125958140686789703409442266739638
17
14:
1.238011918821370329521393256380699769162349718397910977154298360410914958356149017455712625232550734
17
15:
1.199133100749238505019717022643724059545727589005158116085009291789951741097530974751535791705892685
17
16:
1.548633656323688778831066428710723138548523057752184265788259149038051057577045721209148814478236139
18
1:
1.013510707795753000574157136854424057934657439061343224782243428131379294647558584001653837016257081
18
2:
1.027203940676741833224981587924772044715129060944792727708972299646892745865660239605008794648618954
18
3:
1.041082221634774836422973057489282967614889949603896597714243408780662989746256074265038784218385725
18
4:
1.055147877671695617803578109598464967773658920868069380917064739451373025661437442669898610384310857
18
5:
1.069403966903936179670516958833246239201409487548587201750186900823325432963003790923320135645084539
18
6:
1.083851461957661728860494971900242888995185348639050000303004107744754438418503371130533585239562793
18
7:
1.098497818285606317587475628804685905073454423828063590450803566981653771922758018819291009211007204
18
8:
1.113330935354611468277871813088530332982070471992910343639518174551700343381201762197120449725989708
18
9:
1.128398396699953984661835798843924311925608019079749616650162748173542007927705484513327440187740872
18
10:
1.143566822471374958167957360855670609364788137597090796276554885724777682953416350938999265566443354
18
11:
1.159255399705108946284582257605709357443181048003916217161916874597183902397109796297815508333516219
18
12:
1.174212126944176703827862263011695526445196867827985524453793054863444276344638383176346904053447573
18
13:
1.192329865323044681806074131410308935527852316649828002843428288133835612209540827272581661473230865
18
14:
1.202056582858491758556040267477461071697457100095434820537070334405531486252037199894361086399705989
18
15:
1.240947351993061951617235911747037346244546141048536347651473224276827634330332765371464232900615538
18
16:
1.201027851230016276789213622234719165019716340494066751412808318531027035900768814091960645378927701
18
17:
1.549856671586128493760208047416220750023287921731567537228404726164773719895419588441552393328245053
19
1:
1.012795084094686361956286784688599758090929378866439790850647303330625430820862117842488607940801871
19
2:
1.025753887073469195760159514424234666132765585360945248565461039171983444370815037776908772366065012
19
3:
1.038878484785748335488815680672210668797411516139983061420972622774046901285266095357973360317320172
19
4:
1.052171056165338359822583916075655634055125705561387841378602418740822073208010668023401359516220883
19
5:
1.065633575539968311372352884045424172718730171764613393036538297451518922463315221025243127621832557
19
6:
1.079268748998845092196018308673669686590885162902722887163310470276041817309612122970246131156201541
19
7:
1.093077171219087738120258910514720348286073548314831388548949482525697694231591380564059445066542996
19
8:
1.107065968054412323852456466937328876176877088879105128498546683722735772005729366941608169527189193
19
9:
1.121222550090728768709375609921303666763452345999848210801324927887586973562671783919494601106487715
19
10:
1.135594427597321525682685994637457398809449736050914141143898938703119535414493220885258109253673265
19
11:
1.150046987300261874936334793312699049478506344147875054163530133348868713947277666536341925573598542
19
12:
1.165001301161136228485751960948417237346572971392386756497617980783035618866151513448602984518659261
19
13:
1.179198965665129788675257064733284411334043957062171350073661677736852857284328587871346107318734185
19
14:
1.196548194444541790596049999455676651936170004196874244751645142685081173341588036103305566020473208
19
15:
1.205457573825287997004770109169782942576536974516975918384996503104851745627224099072114870995557173
19
16:
1.243579690560255453410254454398146259330003205523407147515841031498676162977963089437121933713445783
19
17:
1.202725691332128149117343093142436017620335022903318931217483537099831058523984479250278966518861481
19
18:
1.550951766711091598747060881993999095753234919615656298480885713867604794908965371111677359290260209
20
1:
1.012151460448179487765272934092530461675374765717590547910579970988556630865154172598088048253988573
20
2:
1.024450577320432602945862285331367903004873215589162017456759790467202895311992023860916254327518614
20
3:
1.036899151164116852853911722149253004587607610982355240646802065829716787861235301478596309799960716
20
4:
1.049498978952239249196019917365267672322593488061975115648896586122364891402924447487657892134978731
20
5:
1.062251956787073287429300508034250748977622145748993374391904449947449014328241891399805959824700401
20
6:
1.075159769136439238785786653724286488444227576027815922501785568645232882030569584590288017902164573
20
7:
1.088224833635922978135298724116231785555387766374967892737869243168968943086001942095264665520308754
20
8:
1.101447433806821098169057012204223104244689808050722431574435447294348547760763842682334582396936405
20
9:
1.114834429062181396544040183630670516630057562423108274729737522342575135936482181824520157548770484
20
10:
1.128372819987095380978699828890029032199003482064703725927448142583599209150691889064211970168406135
20
11:
1.142110084313784456829296023420028871528519549477562787608660768133345241673858699952174866680401416
20
12:
1.155910526602427590977065295044771349337017824748517302254413614057627333446656499347121105577348972
20
13:
1.170196959325325342137407177114396838362126103942022254430001997101548917506233915243607950783679478
20
14:
1.183705226484723965224662219453595610345551176109841499428592360555157281397321012188279001539564587
20
15:
1.200357449607820715875345915374686955558084345546308009913209955585739468743906905816577834593987683
20
16:
1.208526692491796482304810897951641545381764874288977498491523231258349828240374566449030666709062384
20
17:
1.245953569371325403203413352202367356384924964376081033363690444776234488288422106373626042226300943
20
18:
1.204255799649790259944409740651515132378476675156974533557007076283519151326494932026036828517968647
20
19:
1.551938014000424090355718067181258069138383837386206728909292933859988979921435616299258232723092822
Definition
$C(n,k)$ is the least constant in $\|f^{(k)}\|_\infty\leq C(n,k)\|f\|_\infty^{1-k/n}\|f^{(n)}\|_\infty^{k/n}$ for real-valued $f$ on $\mathbb R$ with $f^{(n-1)}$ locally absolutely continuous and $f,f^{(n)}$ essentially bounded; $\|\cdot\|_\infty$ is the essential supremum on $\mathbb R$ [2].
Parameters
$n$
—   highest derivative order ($n\geq2$)
$k$
—   intermediate derivative order ($1\leq k<n$)
Formulas
(1)
$\displaystyle C(n,k)=\frac{K_{n-k}}{K_n^{(n-k)/n}}$, where $\displaystyle K_r=\frac4\pi\sum_{j=0}^{\infty}\frac{(-1)^{j(r+1)}}{(2j+1)^{r+1}}$ for $r\geq0$, with $K_0=1$. Every fractional power is the positive real power [3] [1].
(2)
Let $E_r(x)$ be the Euler polynomial defined by $\frac{2e^{xt}}{e^t+1}=\sum_{r\geq0}E_r(x)t^r/r!$, and put $A_r=|E_r(0)|/r!$ for odd $r$ and $A_r=|E_r(1/2)|/r!$ for even $r$. Then $K_r=\pi^r A_r$, so $\displaystyle C(n,k)^n=\frac{A_{n-k}^{\,n}}{A_n^{\,n-k}}\in\mathbb Q_{>0}$ [4].
(3)
$C(2,1)=\sqrt2$, $C(3,1)=(9/8)^{1/3}$ and $C(3,2)=3^{1/3}$. Also $C(4,1)=(512/375)^{1/4}$, $C(4,2)=\sqrt{6/5}$ and $C(4,3)=(24/5)^{1/4}$ [3].
Comments
(4)
The domain is the entire real line and all three norms are $L^\infty$ norms. On a half-line the corresponding sharp constant already differs when $(n,k)=(2,1)$, where it is $2$ [6].
(5)
The Favard constants are denoted by $K_r$ here; [2] denotes them by $a_r$. They are also called the Akhiezer–Krein–Favard constants.
(6)
Equality is attained by the Euler splines, whose polynomial pieces are scaled Euler polynomials [6]. Their norms give the exact algebraic expression in Formula (2).
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

R = RealBallField(400)
def K(r):
    if r == 0:
        return R(1)
    s = r + 1
    if r % 2:
        return 4/R.pi() * (1 - R(2)**(-s)) * R(s).zeta()
    return (4/R.pi() * R(4)**(-s)
            * (R(s).zeta(R(QQ(1)/4)) - R(s).zeta(R(QQ(3)/4))))

def C(n, k):
    if not 1 <= k < n:
        raise ValueError("Require 1 <= k < n")
    return K(n-k) / K(n)**(QQ(n-k)/n)

print(C(3, 1))
References
[1]
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, §3.3, pp. 212–216.
Links
Similar tables
Euler polynomials —   their antiperiodic extensions form the extremizing splines, and their rational values determine the powers of these constants
Sharp Gagliardo–Nirenberg constants of Del Pino and Dolbeault —   sharp interpolation constants for a different relation between norms of functions and derivatives
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every pair $2\leq n\leq20$ and $1\leq k<n$, covering all intermediate derivatives through order twenty, from the classical second-order inequality to moderate derivative orders)
How they were obtained:

Each value is computed in Arb ball arithmetic with 64 guard bits beyond the 100 significant digits stored. The Favard sums are evaluated through the Riemann zeta function and the Hurwitz zeta function, avoiding an unbounded truncation of the slowly convergent series. At 397 working bits, every computed radius is less than $10^{-110}$.

more

An independent check constructs $E_r(x)/r!$ by exact rational integration and uses Sturm sequences to locate all interior critical points on $[0,1]$. The resulting rational norms determine $C(n,k)^n$ exactly for all 190 entries. The polynomials are compared with the stored Euler polynomials for $0\leq r\leq20$, the classical small cases are checked separately, and the displayed digits of [5] are checked for $C(3,1)$. Exact numerator and denominator root tests establish that no entry in this range is rational. The attached checks also verify each decimal interval read from the database by raising its endpoints to the $n$th power and enclosing the independent rational value.



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