Contents

1 .   (CBS) -- Type Inequalities        1

{1.1}    (CBS)-Inequality for Real Numbers    {1}
{1.2}    (CBS)-Inequality for Complex Numbers     {2}
{1.3}    An Additive Generalisation     {4}
{1.4}    A Related Additive Inequality     {7}
{1.5}    A Parameter Additive Inequality     {8}
{1.6}    A Generalisation Provided by Young's Inequality     {10}
{1.7}    Further Generalisations via Young's Inequality     {12}
{1.8}    A Generalisation Involving J-Convex Functions     {18}
{1.9}    A Functional Generalisation     {20}
{1.10}    A Generalisation for Power Series     {23}
{1.11}    A Generalisation of Callebaut's Inequality     {25}
{1.12}    Wagner's Inequality for Real Numbers     {27}
{1.13}    Wagner's inequality for Complex Numbers     {29}

2.    Refinements of the (CBS)-Inequality         35

{2.1}    A Refinement in Terms of Moduli     {35}
{2.2}    A Refinement for a Sequence Whose Norm is One     {38}
{2.3}    A Second Refinement in Terms of Moduli     {41}
{2.4}    A Refinement for a Sequence Less than the Weights     {44}
{2.5}    A Conditional Inequality Providing a Refinement     {47}
{2.6}    A Refinement for Non-Constant Sequences     {50}
{2.7}    De Bruijn's Inequality     {55}
{2.8}    McLaughlin's Inequality     {57}
{2.9}    A Refinement due to Daykin-Eliezer-Carlitz     {58}
{2.10}    A Refinement via Dunkl-Williams' Inequality     {60}
{2.11}    Some Refinements due to Alzer and Zheng     {61}

3.    Functional Properties         71

{3.1}    A Monotonicity Property     {71}
{3.2}    A Superadditivity Property in Terms of Weights     {73}
{3.3}    The Superadditivity as an Index Set Mapping     {75}
{3.4}    Strong Superadditivity in Terms of Weights     {78}
{3.5}    Strong Superadditivity as an Index Set Mapping     {80}
{3.6}    Another Superadditivity Property     {83}
{3.7}    The Case of Index Set Mapping     {86}
{3.8}    Supermultiplicity in Terms of Weights     {89}
{3.9}    Supermultiplicity as an Index Set Mapping     {93}

4.   Counterpart Inequalities             101

{4.1}    The Cassels' Inequality     {101}
{4.2}    The Pólya-Szegö Inequality     {104}
{4.3}    The Greub-Rheinboldt Inequality    {106}
{4.4}    A Cassels' Type Inequality for Complex Numbers    {107}
{4.5}    A Counterpart Inequality for Real Numbers    {110}
{4.6}    A Counterpart Inequality for Complex Numbers    {113}
{4.7}    Shisha-Mond Type Inequalities    {116}
{4.8}    Zagier Type Inequalities    {118}
{4.9}    A Counterpart in Terms of the sup-Norm {121}
{4.10}    A Counterpart in Terms of the $1-$Norm    {124}
{4.11}    A Counterpart in Terms of the $p-$Norm    {127}
{4.12}    A Counterpart Via an Andrica-Badea Result    {130}
{4.13}    A Refinement of Cassels' Inequality    {133}
{4.14}    Two Counterparts Via Diaz-Metcalf Results    {137}
{4.15}    Some Counterparts Via the Cebysev Functional    {140}
{4.16}    Another Counterpart via a Grüss Type Result    {147}

5.    Related Inequalities            155

{5.1}    Ostrowski's Inequality for Real Sequences    {155}
{5.2}    Ostrowski's Inequality for Complex Sequences    {156}
{5.3}    Another Ostrowski's Inequality    {159}
{5.4}    Fan and Todd Inequalities    {161}
{5.5}    Some Results for Asynchronous Sequences    {163}
{5.6}    An Inequality via A-G-H Mean Inequality    {164}
{5.7}    A Related Result via Jensen's Inequality for Power Functions    {166}
{5.8}    Inequalities Derived from the Double Sums Case    {167}
{5.9}    A Functional Generalisation for Double Sums    {169}
{5.10}    A (CBS)-Type Result for Lipschitzian Functions    {171}
{5.11}    An Inequality via Jensen's Discrete Inequality    {173}
{5.12}    An Inequality via Lah-Ribaric Inequality    {174}
{5.13}    An Inequality via Dragomir-Ionescu Inequality    {176}
{5.14}    An Inequality via a Refinement of Jensen's Inequality    {178}
{5.15}    Another Refinement via Jensen's Inequality    {182}
{5.16}    An Inequality via Slater's Result    {185}
{5.17}    An Inequality via an Andrica-Rasa Result    {187}
{5.18}    An Inequality via Jensen's Result for Double Sums    {190}
{5.19}    Some Inequalities for the Cebysev Functional    {192}
{5.20}    Other Inequalities for the Cebysev Functional    {195}
{5.21}    Bounds for the Cebysev Functional    {197}