Nimertis Mauridis(Math)

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1 ΚΕΦΑΛΑΙΟ 1 ο : ΕΙΣΑΓΩΓΙΚΑ ΣΤΟΙΧΕΙΑ ΓΙΑ ΤΙΣ ΣΥΝΑΡΤΗΣΕΙΣ BESSEL 1.1. ΔΙΑΦΟΡΙΚΗ ΕΞΙΣΩΣΗ BESSEL ΚΑΙ Η ΓΕΝΙΚΗ ΛΥΣΗ ΤΗΣ – ΟΡΙΣΜΟΣ ΣΥΝΑΡΤΗΣΕΩΝ BESSEL 1 ου ΚΑΙ 2 ου ΕΙΔΟΥΣ Η μη ομαλή γραμμική Συνήθης Διαφορική Εξίσωση: καλείται Διαφορική Εξίσωση Βessel και οι αναλυτικές (γραμμικώς ανεξάρτητες) λύσεις της και ονομάζονται συναρτήσεις Bessel 1 ου και 2 ου είδους, αντίστοιχα, τάξεως . Η συνάρτηση παριστάνεται υπό μορφή δυναμοσειράς [65] ως εξής: και η συνάρτηση ορίζεται [65] ως εξής: Η γενική μορφή της λύσης της (1.1) δίνεται από τη σχέση: όπου και σταθερές.

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Nimertis Mauridis(Math)

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  • 1

    1:

    BESSEL

    1.1. BESSEL

    BESSEL

    1

    2

    :

    essel (

    ) Bessel

    1 2 , , .

    [65] :

    [65] :

    (1.1) :

    .

  • 2

    1.2.

    BESSEL

    Bessel :

    :

    .

    , , (1.5) (1.1)

    (1.6) (1.4).

    1.3.

    BESSEL

    BESSEL

    1

    2

    Bessel,

    , :

    (1.5) , .

    (1.7) ,

    Bessel 1 2

    , , .

    (1.7) :

    .

  • 3

    Bessel 1

    [65] :

    :

    ,

    ,

    , Pochhammer.

    Bessel 1

    Bessel 1 :

    :

    Bessel 2 (

    MacDonald) , [65] :

    :

    1.4.

    Bessel 1 2

    [65]:

  • 4

    .

  • 5

    1.5.

    BESSEL

    , : ,

    , , , ,

    , , /,

    [18,19,22,25,34,43,53,54,55,59,64].

    ,

    Rosenthal (1962) [54] Ross (1970) [55]

    .

    ,

    Penfold (2006) [49].

    Q- Marcum,

    radar [10,16,17].

    ,

    ,

    (1984),

    .

    Gronwall (1932) [26]

    . ,

    [57]. ,

    .

    ,

    .

    ,

    .

  • 6

    ,

    Riccatti.

    ,

    Turn

    Bessel.

    Turn

    Bessel

    .

    1 -

    (DDEs) Bessel

    ,

    Riccati

    .

    , .

    , -

    .

    , . ,

    Turn

    . ,

    :

    .

    1.6.

    (

    ) [2,3,28,44,67]

    :

  • 7

    :

    (Completely

    Monotonic)

    ,

    (Strictly Completely Monotonic)

    ,

    (Absolutely Monotonic)

    ,

    (Strictly Absolutely Monotonic)

    , .

    ( )

    (Log - Convex), (Convex), :

    .

    (Convex)

    .

    :

    .

    ( )

    (Log - Concave), (Concave),

    :

    .

    (Concave) , .

  • 8

    :

    T ()

    .

    A

    ,

    .

    M

    .

    , :

    (Berstein - Widder) [67]:

    :

    ,

    .

  • 9

    2:

    BESSEL 1

    2.1.

    (2.1) Baricz (2010) [12].

    ,

    (1.9). ,

    Paris (1984) [48]

    , Robert

    (1990) [53] , Joshi Bissu (1991) [32]

    . , (2.1)

    , .

    :

    H

    ,

    .

    (2.2) Baricz (2010) [12]

    Bordelon [20] Ross

    (1973) [56] Laforgia (1991) [36]

    , ,

    (1.9). (2.2)

  • 10

    .

    :

    , .

    (2.3) Laforgia (1991) [36]

    Soni (1965) [62]:

    (2.3) Baricz (2010) [12]

    ,

    Gronwall (1932) [26]:

    (2.4) Baricz (2010) [12]

    (2.3),

    Gronwall [26].

    . (2.4)

    (2.3).

  • 11

    (2.5) Baricz Sun (2010) [17]

    .

    (2.5)

    (2.3), (2.5)

    ,

    (2.3).

    (2.6) Baricz (2009) [10]

    .

    ,

    (2.6)

    (2.3). , (2.6)

    (2.5).

    -

    Bessel 1 .

    (2.7) (1990)

    [29]

    (2.1).

  • 12

    H (2.8) Joshi Bissu (1996) [33]

    .

    (2.8)

    (2.1) .

    (2.9) Sitnik (1995) [60].

    Sitnik (2.9)

    (2.1).

    :

    , (2.7)

    (2.9).

    (2.10) Baricz (2010) [12]

    .

    ,

    (2.10) (2.1).

    (2.11) Baricz (2010) [12]

    .

    , (2.11)

    (2.1).

  • 13

    :

    (2.10) (2.11),

    , (2.10)

    (2.11).

    (2.12) Baricz (2010) [12]

    . ,

    (2.12) (2.3).

    ,

    , (2.12)

    (2.4)

    , (2.12)

    (2.5). ,

    , (2.6)

    (2.12).

    :

    , (2.6) (2.12)

    , ,

    . (2.12)

    , .

  • 14

    2.2.

    (2.13) Laforgia (1991) [36].

    Soni [62]:

    Soni (1965) Nasell (1974) [45],

    . , (2.13)

    Bordelon [20] Ross (1973) [56]

    , Paris (1984) [48] Joshi Bissu (1991)

    [32]

    .

    H (2.14) Joshi Bissu (1991) [32]

    Nasell (1974) [45]:

    Soni [62],

    . , (2.14)

    (2.13).

    (2.15) Baricz (2009) [9]

    ,

    ,

  • 15

    .

    (2.15) (2.13)

    .

    (2.16) Baricz (2010) [12]

    ,

    .

    (2.16)

    (2.13).

    H (2.17) Joshi Bissu (1996) [33]

    .

    (2.17)

    (2.13), .

    (2.18) Sitnik (1995) [60].

  • 16

    (2.19) Baricz (2010) [12]

    .

    :

    (2.15) (2.19)

    .

    (2.20) Baricz (2010) [12]

    .

    , (2.20)

    (2.13).

    :

    (2.19) (2.20),

    ,

    , (2.20)

    (2.19).

  • 17

    2.3.

    H (2.21) Soni (1965) [62]

    Nasell (1974) [45]

    . ,

    Gupta (1966) [27], De Sitter

    Goovaerts (1972) [61] , Lorch (1967)

    [41] Whittaker. ,

    (2.21) Cochran (1967) [21]

    Jones [31] Reudink (1968) [52]. , Jones

    (1968) :

    Cochran (1967) Reudink (1968) :

    :

    Joshi/Bissu Laforgia (2.21)

    Soni (2.13)

    , .

    (2.22) Nasell (1974) [45]

    (2.21) Soni [62].

  • 18

    (2.23)

    (1990) [29] Mittag-Leffler

    Bessel 1 [65]:

    Bessel 1

    [65]:

    (2.24) (1991)

    [30] Baricz (2010) [12]

    .

    , (2.24)

    (2.21) Soni.

    (2.25) Baricz (2010) [12]

    Bessel 1 . ,

    , :

  • 19

    , (2.25)

    (2.24).

    :

    (2.24) (2.25)

    ,

    , .

    (2.26) (2012) [35]

    (1.15.iii) (1.15.iv)

    (

    ).

    Bessel 1

    .

    (2.27) (2012) [35]

    ( ) :

    ,

    ,

    (2.27)

    (2.24),

    , .

  • 20

    (2.28) (2012) [35]

    ( )

    [65]:

    ,

    :

    (2.28) (2.24),

    .

    (2.29) (2012) [35]

    (

    ).

    :

    (2.29) (2.28),

    .

  • 21

    ,

    Bessel 1 .

    (2.30) (2012) [35]

    .

    (2.31) Segura (2011) [57].

    ,

    :

    ,

    :

    :

    (2.31) :

  • 22

    .

    (2.32) Segura (2011) [57]

    :

    , (2.31).

    (2.32)

    .

    2.4.

    Bessel 1

    .

    (2.33)

    (1990) [29] Mittag-Leffler

    Bessel 1 [64]:

  • 23

    Bessel 1

    :

    (2.34) Baricz (2010) [12]

    , .

    (2.35) Baricz (2010) [12]

    , . ,

    (2012) [35]

    ( ,

    ).

    (2.36) (2012) [35]

    (1.15.iii) (1.15.iv),

    (

    ), Mittag-Leffler

    Bessel 1 [65]:

  • 24

    [65]:

    :

    Bessel 1

    . (2.34)

    Laforgia Natalini (2010) [40] Segura (2011) [57].

    (2.37) (2012) [35],

    (2.27)

    (

    ) [65]:

    ,

    ,

    (2.37)

    (2.34),

    , .

    (2.38) (2012) [35]

    , (2.28),

  • 25

    (

    ) [65]:

    ,

    :

    (2.38) (2.34),

    .

    (2.39) (2012) [35]

    , (2.29),

    (

    ).

    :

    (2.39) (2.38),

    .

    ,

    Bessel 1 .

    (2.40) (2012) [35]

    . ,

  • 26

    ,

    . [65]:

    :

    :

    (2012) [35] ,

    :

    , .

    ,

    , ,

    . ,

    :

    (2.41) (2.42) Segura (2011)

    [57]

    (2.31).

    :

  • 27

    ,

    .

    :

    (2.42)

    (2.41), :

    ,

    .

    2.5.

    (2.43) (2012) [35]

    (

    ), (1.15.iv),

    Bessel (1.7). ,

    (2.43) Segura (2011) [57]

    (2.31).

  • 28

    (2.44) Segura (2011) [57]

    :

    (2.31).

    (2.44)

    . :

    , (2.44) Gronwall

    (1932) [26], . , Phillips

    Malin (1950) [51] , . ,

    (2.44) Amos (1974) [4],

    .

    :

    (1.15.iii) (1.15.iv),

    (2.44)

    Turn:

    (2.45) Segura (2011) [57]

    :

  • 29

    (2.31).

    (2.45)

    . :

    (2.45) Amos [4],

    .

    2.6.

    (2.46) Gronwall (1932) [26].

    :

    (2.46) Gronwall Laforgia

    (2.4),

    .

    :

    (2.5)

    Gronwall,

    .

  • 30

    (2.47) Baricz Sun (2009) [16]

    Gronwall [26] Nasell [45]

    (2.6).

    :

    (2.46) (2.47)

    ,

    (2.47) .

    (2.48) (2012) [35]

    (1.15.iv)

    ,

    .

    (2.48)

    , , .

    (2.49) (2.50) Segura (2011)

    [57] :

    (2.31).

  • 31

    ,

    , .

    (2.49) (2012)

    [35]

    (

    ), (1.15.iv),

    Bessel (1.7).

    Baricz (2009) [9] Laforgia/Natalini (2010) [40]

    .

    :

    (2.50)

    ,

    .

    (2.51) Segura (2011) [57]

    :

    Turn:

    Bessel 1 .

    (2.52) Phillips Malin (1950) [51]

    Turn:

  • 32

    ,

    Thiruvenkatachar Nanjundiah (1951), Amos (1974) [4],

    Joshi/Bissu (1991) [32] Baricz (2010)

    [12].

    :

    (2.49)

    (2.52)

    (2.53) Segura (2011) [57]

    :

    (2.31).

    (2.53) ,

    :

    .

  • 33

    2.7.

    (2.54) Segura (2011) [57]

    (1.15.iii) (1.15.iv)

    Riccatti:

    (2.54).

    :

    , :

    (2.54),

    .

  • 34

    2.8.

    (2.55) Laforgia (1991) [36]

    (2.3) (2.13).

    (2.56) Joshi Bissu (1996) [33]

    (1.15.iv)

    (2.55)

    .

    (2.3).

    (2.57) Baricz (2010) [12]

    (2.4) (2.19). T

    (2.57)

    (2.55),

    .

  • 35

    :

    A

    , (2.56) (2.57)

    .

    (2.58) Baricz (2010) [12]

    (2.4) (2.20).

    , (2.58) (2.57).

    :

    A

    , (2.56) (2.58)

    .

    (2.59) Baricz (2010) [12]

    (2.6) (2.15)

    . , (2.59)

    (2.55).

  • 36

    ,

    (2.59)

    Laforgia (2.56).

    :

    A

    , (2.56) (2.59)

    .

    2.9.

    (2.60) Laforgia (1991) [36]

    (2.3) (2.13).

    (2.61) Joshi Bissu (1996) [33]

    (1.15.iv)

    .

  • 37

    :

    O Joshi Bissu

    ,

    (2.61)

    (2.60). .

    .

    . ,

    ,

    . ,

    (2.61) .

    (2.62) Baricz (2010) [12]

    (2.4) (2.19).

    , (2.62)

    (2.60).

    (2.63) Baricz (2010) [12]

    (2.4) (2.20)

    (2.62).

  • 38

    (2.64) Baricz (2010) [12]

    (2.15) (2.19).

    (2.60). , ,

    . , (2.64)

    (2.62).

    2.10. O TURAN

    (2.65) Baricz (2009) [6]

    ( ) .

    .

    (2.65) Thiruvenkatachar Nanjundiah

    (1951) [63], Amos (1974) [4] Joshi/Bissu (1991)

    [32]. , Segura (2011) [57]

    (2.31).

    :

    , Lorch (1994) [42] (2.65)

    Turn (generalized

    Turnian):

    Turn (2.65) .

  • 39

    (2.66) ,

    Thiruvenkatachar Nanjundiah (1951) [63], Amos (1974) [4],

    Joshi Bissu (1991) [32] Baricz

    (2010) [12] Segura (2011) [57].

    :

    ,

    Turn (2.66) .

    (2.67) Baricz (2010) [12]

    , .

    (2.68) Baricz (2010) [12]

    , .

    Segura (2011) [57] ,

    , (2.31)

  • 40

    Bessel 1 .

    . (2.65)

    (2.66)

    .

  • 41

    3:

    BESSEL 2

    3.1.

    (3.1) Laforgia (1991) [36]

    .

    Ross (1970) [55],

    Bordelon (1973) [20] Paris (1984) [48].

    :

    ,

    ,

    .

    H (3.2) Laforgia (1991) [36]

    o

    , ,

    . , (3.2)

    (3.1).

  • 42

    (3.3) Joshi Bissu (1991) [32]

    (2.1)

    ,

    .

    (3.3)

    (3.1).

    :

    H

    , ,

    .

    (3.4) Joshi Bissu (1991) [32]

    ,

    .

    (3.5) Laforgia (1991) [36]

    ,

    (3.1) (3.2). (3.1)

    .

    ,

    ,

  • 43

    . , ,

    (3.5)

    .

    :

    ,

    ,

    ,

    .

    (3.6) Laforgia (1991) [36]

    Soni

    Bessel 2 :

    .

    (3.7) Joshi Bissu (1996) [33].

    , (3.5)

    (3.7).

    :

    Bessel 2 :

  • 44

    (3.8) Baricz (2010) [12]

    ,

    . ,

    Weniger Cizek (1989) [66]

    [24]:

    .

    ,

    (3.8) (3.1),

    .

    H (3.9) (3.8).

    (3.10) Miller Samko (2001) [44]

    .

    (3.10)

    Laforgia (3.5),

    . ,

    , (3.10)

    (3.5) (2.5)

    Bessel 1 .

    ,

    (3.10)

    (3.8).

  • 45

    :

    :

    .

    3.2.

    (3.11) ,

    Bessel 1 Soni (1965)

    [62].

    (3.12) Segura (2011) [57]

    .

    ,

    , :

    .

  • 46

    (3.13) - (3.16) Segura (2011) [57]

    :

    :

    (3.13) (3.14)

    , ,

    (3.15) (3.16)

    , .

    (3.13) ,

    .

    (3.14)

    ,

    (3.15)

    .

  • 47

    :

    , ,

    ,

    .

    :

    ,

    .

    (3.17) Segura (2011) [57]

    Bessel 2 .

    :

    (3.16)

    (3.17).

  • 48

    3.3.

    (3.18) Segura (2011) [57],

    (1.16.iii) (1.16.iv)

    Riccatti:

    (3.18).

    (3.18).

    (3.19) (3.18).

  • 49

    3.4.

    (3.20) Segura (2011) [57]

    :

    (3.13) (3.16).

    (3.22)

    .

    (3.21) Segura (2011) [57]

    :

    (3.13) (3.16).

    (3.21)

    .

  • 50

    :

    (3.20)

    (3.21),

    (3.20) .

    (3.22) Segura (2011) [57]

    :

    (3.13) (3.16).

    (3.22)

    .

    (3.23) Segura (2011) [57]

    :

    (3.13) (3.16).

    (3.23)

    .

  • 51

    (3.24) Segura (2011) [57]

    Bessel 2

    .

    :

    (3.23)

    (3.24).

    3.5. O TURAN

    Segura (2011) [57]

    , ,

    Bessel 1

    , .

    (3.26) Segura (2011) [57]

    ,

    :

  • 52

    ,

    ,

    , (3.26) .

    :

    Turn :

    .

    E Turn

    . , .

    :

    ,

    .

    (3.27) Segura (2011) [57]

    :

    . :

    ,

  • 53

    :

    , (3.27)

    .

    (3.28) Segura (2011) [57]. ,

    . :

    ,

    . ,

    , :

    (3.29) Segura (2011) [57]

    (3.28),

  • 54

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    [7] A. BARICZ, Turn-type inequalities for hypergeometric functions, Proc.

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    [8] A. BARICZ, Functional inequalities involving Bessel and modified Bessel

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    [11] A. BARICZ, Turn-type inequalities for some probability density functions,

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    [12] A. BARICZ, Bounds for modified Bessel functions of the first and second

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  • 55

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    [17] A. BARICZ AND Y. SUN, Bounds for the generalized Marcum Q-function,

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